Limit of cosØ-1 over Ø as Ø approaches 0 is 0 I Special limit of Trigonometric Functions

Summary

This video script delves into the process of finding the limit of a trigonometric function as Theta approaches zero. It explores the indeterminate form issue when directly substituting zero, then employs the conjugation method to rationalize the expression. The script explains the use of trigonometric identities, specifically the Pythagorean identity, to simplify the problem. It also covers the limit of a product involving sine and tangent functions, illustrating the transformation of tangent into sine and cosine, and the application of limit theorems to reach the conclusion that the limit is one. The detailed explanation aims to guide viewers through the intricacies of limit calculations in trigonometry.

Takeaways

• 📚 The focus of the lesson is to find the limit of a trigonometric function, specifically cosine Theta minus 1 divided by Theta as Theta approaches zero.
• 🚫 Direct substitution results in an indeterminate form (0/0), which indicates that the limit does not exist without further analysis.
• 🔍 The method of conjugation is introduced to rationalize the expression, which involves multiplying the numerator and denominator by the conjugate of the numerator.
• 🔢 The Pythagorean trigonometric identity is used, where sin squared Theta plus cosine squared Theta equals 1, to simplify the expression.
• 🔄 The process involves transmuting the expression to involve sine squared Theta, which is equivalent to 1 minus cosine squared Theta, to facilitate simplification.
• 📉 The limit of sine squared Theta over Theta as Theta approaches zero is used, which is a key step in solving the limit.
• 🔑 The limit of Theta over sine Theta as Theta approaches zero is identified as a crucial theorem, which states that this limit is equal to one.
• 📌 Factoring is used to simplify the expression, allowing for the separation of terms and the application of limits to each factor individually.
• 🎯 The final result of the limit is determined to be zero after applying the above methods and simplifications.
• 📝 The script also discusses another limit involving the product of Theta and sine Theta over the square of tangent Theta as Theta approaches zero, which is transformed into sine squared Theta over cosine squared Theta.
• 🧩 The final limit of the second problem is found to be one, after simplifying and applying the limit of the ratio of Theta to sine Theta as Theta approaches zero.

Q & A

What is the main topic discussed in the video script?

The main topic discussed in the video script is finding the limit of a trigonometric function, specifically the limit of (cosine Theta - 1) / Theta as Theta approaches zero.

Why is direct substitution not applicable when Theta approaches zero in the given problem?

Direct substitution is not applicable because it results in an indeterminate form of 0/0, which means the limit cannot be determined by simply substituting Theta with zero.

What method is suggested to resolve the indeterminate form of 0/0 in the problem?

The method of conjugation or rationalization is suggested to resolve the indeterminate form. This involves multiplying the numerator and the denominator by the conjugate of the numerator.

What is the conjugate of the numerator in the given problem?

The conjugate of the numerator, which is cosine Theta - 1, is cosine Theta + 1.

How does the Pythagorean trigonometric identity help in solving the problem?

The Pythagorean trigonometric identity, which states that sin^2(Theta) + cos^2(Theta) = 1, helps in transforming the expression into a form that can be simplified and factored to find the limit.

What is the significance of the limit of sin(Theta)/Theta as Theta approaches zero in the script?

The limit of sin(Theta)/Theta as Theta approaches zero is significant because it is equal to 1. This fact is used to simplify the expression and find the limit of the given trigonometric function.

What is the final result of the limit of (cosine Theta - 1) / Theta as Theta approaches zero?

The final result of the limit of (cosine Theta - 1) / Theta as Theta approaches zero is zero.

What is the second problem discussed in the script related to the limit of a trigonometric function?

The second problem discussed is finding the limit of (Theta * sin(Theta)) / (tangent(Theta)^2) as Theta approaches zero.

How is the tangent function in the second problem converted to sine and cosine functions?

The tangent function is converted by expressing it as the sine function squared over the cosine function squared, i.e., sin^2(Theta) / cos^2(Theta).

What is the final result of the limit of (Theta * sin(Theta)) / (tangent(Theta)^2) as Theta approaches zero?

The final result of the limit of (Theta * sin(Theta)) / (tangent(Theta)^2) as Theta approaches zero is one.

Why is it important to rationalize the expression in the script?

Rationalizing the expression is important to eliminate the complex or indeterminate forms, allowing for a clearer path to finding the limit of the given trigonometric function.

Outlines

00:00

🧭 Calculating Indeterminate Limits with Trigonometric Functions

The paragraph discusses the process of finding the limit of a trigonometric function as Theta approaches zero. The specific function is cosine Theta minus 1 divided by Theta. Direct substitution results in an indeterminate form of 0/0. To resolve this, the method of conjugation is introduced, multiplying the numerator and denominator by the conjugate of the numerator, which is cosine Theta plus 1. This leads to a rationalized form that utilizes the Pythagorean trigonometric identity, simplifying the expression and allowing for the determination of the limit as zero.

05:01

🔍 Applying Trigonometric Identities to Simplify Limits

This section delves into the use of trigonometric identities to simplify the limit of sine squared Theta plus cosine squared Theta, which equals 1. The process involves transmuting the expression to isolate terms that can be substituted with known limits, such as sine squared Theta over negative 1, which simplifies to cosine squared Theta. The limit of sine Theta over Theta as Theta approaches zero is then used to simplify the expression further, ultimately leading to the conclusion that the limit of the original function is zero.

10:01

📉 Factoring and Simplifying Limits of Trigonometric Functions

The paragraph describes the process of factoring and simplifying the limit of a function as Theta approaches zero. It involves recognizing the Pythagorean identity in the numerator and applying the limit of sine Theta over Theta, which is one, to factor out terms. The resulting expression is simplified by direct substitution, leading to the conclusion that the limit of the function is zero. The explanation emphasizes the importance of considering multiple steps and theorems in the process of finding the limit.

15:03

📚 Transforming Tangent to Sine and Cosine for Limit Calculation

This part of the script focuses on transforming the tangent function into sine and cosine to simplify the limit calculation as Theta approaches zero. The limit of the product of Theta and sine Theta over tangent squared Theta is converted into sine squared Theta over cosine squared Theta. The process involves recognizing patterns that can be simplified using known limits and trigonometric identities, ultimately leading to the conclusion that the limit of the given function is one.

20:03

🔄 Further Exploration of Limit Transformations and Simplifications

The final paragraph continues the exploration of limit transformations, specifically focusing on the limit of the product of Theta sine Theta over tangent squared Theta as Theta approaches zero. The paragraph discusses the cancellation of terms and the application of the limit of Theta over sine Theta, which is one, to simplify the expression. The limit of cosine squared Theta as Theta approaches zero is also used, leading to the final conclusion that the limit of the original function is one.

Keywords

💡Limit

In the context of the video, 'limit' refers to the value that a function or sequence approaches as the input of the function approaches some value. It is a fundamental concept in calculus and is central to understanding the behavior of functions, especially as variables approach infinity or specific values. The script discusses finding the limit of a trigonometric function as Theta approaches zero, illustrating the process of evaluating limits through direct substitution and other methods.

💡Trigonometric Function

A 'trigonometric function' relates angles to the ratios of the sides of a right triangle or to the coordinates of points on the unit circle. The video script specifically discusses the cosine function, which is the ratio of the adjacent side to the hypotenuse in a right triangle. The limit of the cosine function as Theta approaches zero is a key point in the video, demonstrating the application of trigonometric functions in evaluating limits.

💡Direct Substitution

Direct substitution is a method used in calculus to find the limit of a function by simply substituting the value that the variable approaches. In the script, the instructor attempts to find the limit of cos(Theta) - 1 / Theta by directly substituting Theta with zero, leading to an indeterminate form, which then prompts the use of other methods to find the limit.

💡Indeterminate Form

An 'indeterminate form' occurs when a limit calculation results in an expression like 0/0 or undefined, which does not provide a clear limit value. In the video, the direct substitution of Theta with zero in the expression cos(Theta) - 1 / Theta results in an indeterminate form, indicating that an alternative approach is needed to determine the limit.

💡Conjugation

In the context of the video, 'conjugation' refers to the process of multiplying the numerator and the denominator of a fraction by the conjugate of the numerator to simplify the expression and help find the limit. The script mentions using the conjugate of cos(Theta) - 1, which is cos(Theta) + 1, to rationalize the expression and avoid the indeterminate form.

💡Rationalization

'Rationalization' is a mathematical technique used to eliminate square roots or other radicals from a denominator, often by multiplying by a form of one that includes the conjugate. In the script, rationalization is used to transform the expression involving the cosine function into a form that allows the limit to be determined as Theta approaches zero.

💡Pythagorean Identity

A 'Pythagorean identity' in trigonometry is an equation that holds true for sine and cosine functions, such as sin^2(Theta) + cos^2(Theta) = 1. The script refers to the Pythagorean identity to simplify the expression involving the cosine function and to help find the limit by showing that sin^2(Theta) is equivalent to 1 - cos^2(Theta).

💡Substitution Method

The 'substitution method' involves replacing a part of an expression with another equivalent expression to simplify the problem. In the video, the substitution method is used to replace cos^2(Theta) - 1 with -sin^2(Theta), which is derived from the Pythagorean identity, to help find the limit of the given function.

💡Factoring

'Factoring' is the process of breaking down a polynomial or expression into a product of its factors. In the script, factoring is used to simplify the expression after applying the Pythagorean identity and substitution method, which helps in isolating terms that can be evaluated as Theta approaches zero.

💡Tangent Function

The 'tangent function' in trigonometry is the ratio of the opposite side to the adjacent side in a right triangle or the sine function divided by the cosine function. The script discusses converting the tangent function into sine and cosine terms to simplify the process of finding the limit as Theta approaches zero.

💡Reciprocal

A 'reciprocal' of a number is the value which, when multiplied by the original number, results in one. In the video, the reciprocal of the tangent function is used to express the given limit problem in terms of sine and cosine, which is a step towards simplifying the expression and finding the limit.

Highlights

Finding the limit of the trigonometric function cos(Θ) - 1 / Θ as Θ approaches zero.

Direct substitution results in an indeterminate form, indicating the need for alternative methods.

Introduction of the conjugate method for rationalizing the expression.

The use of trigonometric identities to simplify the expression, specifically the Pythagorean identity for sine and cosine.

Transformation of the numerator into a difference of squares to facilitate further simplification.

Multiplication of the numerator and denominator by the conjugate of the numerator to rationalize the expression.

Application of the limit theorem that sin(Θ) / Θ approaches one as Θ approaches zero.

Factoring out constants and applying the limit theorem to simplify the expression.

Substitution method used to replace cos²(Θ) - 1 with -2sin²(Θ) for further simplification.

Division of both sides of the trigonometric identity by -1 to align with the expression's numerator.

Final simplification leading to the conclusion that the limit of the original expression is zero.

Transition to solving the limit of the product of Θsin(Θ) over tan²(Θ) as Θ approaches zero.

Conversion of tangent to sine and cosine to simplify the problem.

Use of reciprocal properties to rewrite the expression and simplify the limit calculation.

Cancellation of terms and factoring to isolate the limit of Θ / sin(Θ) as Θ approaches zero.

Application of the limit theorem for Θ / sin(Θ), concluding the limit of the expression is one.

Summary of the process and the importance of considering multiple methods for finding limits in trigonometric functions.

Transcripts

our focus in here is again finding the

limit of trigonometric function okay now

one of our problem is it says in here

find the limit of cosine Theta minus

1 over Theta as your Theta is

approaching zero now so that you will

notice when we make use of the direct

substitution shall we say we're going to

use the direct substitution in here here

we have now the

cosine cosine Theta which is the same as

zero which is the same as 0 -

one and of course that is the same as

Theta is zero now it will give you now a

result of 1 - 1 so cosine 0 is 1 - 1

over Z so it will give you a result

which is the same as 0 over zero which

is an

indeterminate okay indeterminate number

meaning to say if it is in determinate

there will always be the there will

always be the uh limit but of course we

need to look into other processes in

looking for the limit of this problem

okay so if that is undefine then the

limit does not exist but because it is

indeterminate there is the limit okay

okay so how is that now in here we make

use of the method of conjugation or the

rationalization method so we think of a

number uh that is to multiply in here

but it will never change the problem

definitely that is one okay so let's try

to

have the

limit the limit of uh we have in here

con

sign

cine Theta

-1/

Theta as your Theta is

approaching approaching zero now what

would be our

multiplier again as I said one okay our

multiplier is one but remember remember

our one must be defined okay must be

defined so what is the uh definition of

our one here the definition of our one

in here will be the conjugate of the

numerator what is the conjugate that is

the same as uh

cosine okay cosine Theta + 1

over

cine Theta + 1 so that is our definition

for the one so meaning to say uh because

our multiplier is one we change that one

to the this uh this one using uh by

using it as our by using this one to

rationalize this problem so what how is

that that is the same as

saying uh multiplied by

cosine cosine Theta + 1 over

cosine Theta + 1 so actually we are

having the idea of of rationalization

method using the conjugate of the uh

numerator now having said that one we

are now uh going to multiply so multiply

numerator to numerator denominator to

denominator but

notice notice that in our

numerator in our numerator that is

actually the difference of two perfect

square which is the same as cosine squar

okay

cine s Theta - 1 s or

minus-1 well of course the denominator

well let's try first to maintain uh in

their uh product form okay so Theta

multiplied by the

denominator that is the same as

cosine theta plus

+ 1 definitely your limit is uh the

limit your

Theta is

approaching zero okay it is approaching

zero but uh notice that uh in our

numerator the resulting statement for

our numerator is actually a result of

Pythagorean theorem or trigonometric

pythagorean identities which is uh you

have to remember that in our Pythagorean

theorem of the trigonometric

identities the uh uh Sin

squ Sin squ theta

plus

cine squ Theta results to 1 okay again

remember this equations so so remember

note okay let's try to write it

Note and Note again another note or

remember okay so notice that in here

when we are going to uh when we are

going to bring out the numerator in here

we have we have sign okay we have we

have to make ramification of this sin s

Theta is equal

to co

cosine cosine s theta plus + 1 trans uh

that is we have in here the

transmutation okay

or uh transpose the cosine so it will

become negative now but notice that uh

our cosine in here is negative but in

here that is positive same true with the

uh one in here positive that is negative

so what are we going to do what are we

going to do with this resulting uh

resulting statement for sin squ Theta we

have to divide both sides okay divide

both sides by what

by-1 now what is our purpose so that we

will make this one be the same as in our

numerator so if we going to divide this

one

we can have now the

statement uh sin 2 Theta / -1 we

have

sin s Theta okay is equal

to cosine 2 Theta /1 it will become now

cosine squ Theta now what about this one

pos 1 /1 it will give you

-1 notice that upon the upon the

division of -1 this statement

now is the same as our

numerator so meaning to say by

substitution

method okay by substitution method we

can now have a new statement

saying the limit of by

substitution what are we going to use

sin 2 Theta the limit of by substitution

method we can have now the limit of

netive

sin squ Theta so we replace this one uh

with its uh definition what is the

definition of this one negative sin squ

Theta as you can see it is but the same

okay this statement

is the same

[Music]

with this okay let's try to make an

arrow so that you can uh identify so by

substitution we can have now new

statement the limit of sin s Theta over

the same okay

over uh Theta the product of theta

multiplied by

cosine cosine Theta

+ 1 okay cosine Theta + 1

definitely as your

Theta as your Theta is approaching zero

still we cannot uh still

uh uh have the direct substitution in

here so that you will notice again in

this statement we have to remember again

that

uh in one of the theorem the limit

of

s Theta

over

Theta as your Theta is

approaching zero is the same as one so

with this we can we can make

ramification of this one again to at

least bring out this theorem so as you

can see the numerator is in a product

form the denominator is in a product

form and so that we with that we can

have now the factoring okay we can

factor that one so the

limit as your Theta is

approaching zero now what are the factor

the factor now is

NE -1 multiplied by

S

sin

Theta multiplied by sin

Theta okay multipli by sin Theta

definitely

over uh

Theta uh multiplied by that is the same

as that is the same as

cosine uh Theta

+ 1 okay so those are the factor of the

numerator now with that we can separate

again them by

factoring okay so we can have we can

factor out this one by

saying -1 how about in here we have the

limit times the

limit okay we have the limit of we have

the limit of uh s and we have the limit

of sin Theta we Factor out1 so limit of

s

Theta over Theta over Theta as it

approaches as approach as your Theta is

approaching zero now multiplied by

multiplied by another one

s

Theta

over

cosine Theta + 1 okay so we have in here

the equation now notice that

uh we have this

statement this statement now says

that this is the same

as this note so with that we can say now

that we have in here another statement

saying

-1 the result of this one is multiplied

by result is one okay and of course

we can have the direct substitution now

in here that is the same as uh that is

the same as sign s

uh

s sin 0 is the same

as 0 sin 0 is the same as zero

over uh substitute this one

cosine 0 is 1

+ 1 so notice the result is

what -1 M1 that is1 multiplied by 0 the

result is 0 over 2 now dividing that one

the result is zero so that is the that

is the uh limit of our problem so we can

say now we can say now that uh the limit

the limit yeah the limit of

cosine

cosine Theta -

1

over Theta - Theta as your Theta is

approaching zero is

what zero okay so meaning zero the

result will

be zero so notice that uh when we had

when we had the result which is zero

there are so many things that we need to

consider okay so there are so many thing

that we need to consider one uh think of

this that uh we need to uh we need to

think we need to

uh rationalize the problem by getting

the conjugate of our of our numerator

and so how are we going to rationalize

you use the uh the conjugate of this one

as our multiplier okay so meaning to say

why do we

multiply uh so that uh of course we can

reduce or we can change the uh problem

in terms of sign okay so our multiplier

is always the conjugate over the

conjugate or that is the same as one so

that it will not change the problem

now after that one notice that we have

now in here the uh numerator that is the

same as uh the result of

Pythagorean identities of trigonometric

function which is coming from this

statement okay so that with that with

that we make ramification of this

pythagorean identities and so bring out

this numerator cosine s Theta minus 1 by

having this statement okay sin 2 Theta

is the same as cosine 2 Theta minus 1

which is the same as in here with that

by uh by

substitution okay by sub

substitution we change we change the

cosine squ Theta minus 1 2 sin 2 Theta

but uh again notice that this is to be

uh to be factored okay why because if we

do not Factor this one it will result

again to as I said as I said uh in

determinate result so we apply the one

of the theorem that state that the limit

of s Theta over Theta as your Theta is

equal to uh approaches zero is the same

as one so having said that one uh we

Factor this one so that we will bring

out okay we will bring out this the so

factoring sin 2 Theta is the same as -1

Sin Sin Theta * sin Theta definitely the

denominator will always be the same with

that we factor out separately so we

factor out one in here -1 as one of the

property of the uh limit we can factor

out so how about in here we can we can

have the statement limit of sin Theta

over Theta as your Theta is approaching

zero which is the same as that one so

that it will result to one how about the

other Factor we have the in here sin

Theta over cosine Theta + 1 we can have

the direct substitution now in here so

when we substitute this one sin Theta

into sin 0 it will give you zero how

about this one cosine Theta which is

Theta is zero that is that is one + 1 so

it will give you 0/ 2 so dividing it the

result will be zero and so we can

establish now that the limit of the

cosine Theta - 1 / Theta as your T is

approaching zero is the same as zero I

hope you understand the process and

thank you so much uh again please uh do

not uh forget to subscribe and click the

notification Bell so that uh he will

always be notified should there be any

problem or any video Lesson that is

being posted thank you and

more again we have in here the uh

problem uh that is the same as the limit

of the product of theta sin Theta over

tangent squ Theta as your Theta is

approaching zero now how are we going to

solve for the limit of this one now

notice that we have in here tangent and

sign now remember that uh in the

pre-calculus we stated that we always

try to convert or transform the tangent

in terms of cosine or sign okay so with

that we can have the equation by

saying the

limit as your Theta is

approaching

zero

of

theta s

[Music]

Theta

is the same

as is the same as we convert this one

into what uh s and cosine so this is the

same as this is the same

as Sin squ okay sin

s Theta over

cosine squ Theta so that is the

converted form of that one now having

said that one we are now uh we can

continue now our ramification of the

problem we can have now the next

statement saying the

limit as your Theta is approaching zero

will give you a result okay so get the

rece of this one and proceed to

multiplication so that is the same

as Theta

s Theta multiplied by the reciprocal of

this one that is the same as uh

cosine cine s

Theta

over

sin squar Theta okay our sin squ Theta

now now with this now we can have now

the uh multiplication or shall we say

the division we can have the division

okay by how we can have again the other

statement let's right we start here the

limit as your Theta is

approaching zero we can have this

statement by

saying

uh Theta

okay Thea

s

Theta

[Music]

over

over

[Music]

s

Theta multiplied

by

S

Thea we can have the

one okay so of course that is the same

as multiplied by

cosine

squar Theta now notice that uh one of

the statement

sign this one can canceled out can be

canceled out and so we can have now the

statement the

limit as Theta is

approaching zero

and have now the

statement

Theta okay over

s

Theta

[Music]

okay and of

course we have now in

here

cosine

cosine

squar

Theta now with that we can have another

factoring so we can Factor this one so

by

saying to uh bring out the

limit that states that uh the

limit

of theta

over

s over s Theta as your Theta is

approaching zero will always give you

one so because of this we can we have to

factor this things this one okay so we

can have now this statement

saying the limit of

theta over

s Theta as your Theta is

approaching zero okay or shall we say

cosine okay cosine squ Theta

Okay so with that notice that

uh uh if we are going

to yes yes notice that if we are going

to have now the result of this

one of this

one that would be the same as one and

the limit of cosine cosine squ

Theta okay the limit of cosine squ

Theta multiplied

by multiplied

by 1 okay so the limit of cosine squ

Theta by substitution cosine

squ uh Theta which is Theta is zero that

is the same as one so the result will

always be one 1 multip by 1 is one okay

so so we can say now we can say now that

uh the limit okay can say know that the

limit

of of theta sin Theta over tangent squ

Theta as your Theta is

approaching approaching zero will be one

so I hope you got the procedure of

having the result of this

problem thank

you and