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

Dr.Gajendra Purohit
25 Dec 202014:31

Summary

TLDRThis 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.

Outlines

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Keywords

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Transcripts

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Engineering MathFunctionsLimitsDifferentiabilityMathematics BStudent VideosLearning ToolsConcept MappingMath ExamplesEducational Content
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