Real Analysis | Limit of Function - Concept of Limit, Left hand & Right hand Limit

Summary

This video covers essential concepts in engineering mathematics, focusing on the definition of functions and limits. The instructor explains how a function maps elements from one set to another and delves into the limit of a function as a key concept. Examples of limits and their notations are discussed, including one-sided limits and conditions for the existence of limits. Several problem-solving examples related to limits and differentiability are presented. The video concludes with a reminder to subscribe for more content on advanced mathematical topics.

Takeaways

• 📘 The video is for Engineering Mathematics B, focusing on the concept of limits in functions.
• 📐 A function is defined as a rule of mapping where each element in set X is assigned to a unique element in set Y.
• 📊 X represents the domain of a function, and the domain is an important concept in defining functions.
• 🧮 Example of a function: f(x) = x², with values like 1, 2, and their corresponding outputs such as 1, 4, 9, 16, 25.
• 📉 The range of a function is a subset of the codomain, which includes all possible output values.
• 📏 A limit of a function at a certain point is denoted as lim (x → a) f(x).
• ⚖️ The left-hand limit of a function is denoted as lim (x → a⁻) f(x), and the right-hand limit as lim (x → a⁺) f(x).
• ❌ The speaker mentions a specific function where the limit does not exist under certain conditions.
• 🔄 The example provided involves a piecewise function based on whether x is rational or irrational.
• 🔍 The final segment discusses differentiability and logarithmic functions, with an example involving lim (x → 0) of log(1 + x)/x.

Q & A

What is the definition of a function as described in the transcript?

A function is a rule of mapping in which every element of a set X is assigned a unique element in set Y.

What is the domain of a function?

The domain of a function is the set of all possible input values (X) for the function, as explained in the transcript.

What is an example of a function provided in the transcript?

An example of a function given is f(x) = x², where for inputs like 1, 2, 3, the corresponding outputs are 1, 4, 9, respectively.

How is the range of a function defined in the transcript?

The range of a function is defined as a subset of the codomain (Y), consisting of all output values generated by the function.

How is the limit of a function denoted in the transcript?

The limit of a function f(x) as x approaches a is denoted as lim(x -> a) f(x).

What does the term 'left limit' refer to?

The 'left limit' of a function at a point a is the value that f(x) approaches as x approaches a from the left, denoted as lim(x -> a-) f(x).

What is the significance of the limit in terms of function continuity?

The existence of a limit at a point indicates that the function approaches a specific value as x approaches that point. If the left and right limits are equal, the function is continuous at that point.

What is an example of a function with no limit provided in the transcript?

An example provided is f(x) = x if x is rational and -x if x is irrational. The transcript explains that for this function, the limit does not exist at certain points.

What does the transcript mention about solving limits for certain functions?

The transcript suggests that for some functions, such as when x > 1 or x < 1, the limit can be solved more easily.

What is the logarithmic limit example mentioned in the transcript?

The transcript provides an example of a logarithmic limit: lim(x -> 0) log(1 + x) / x.