03_Ex 1.3

Exercise Questions

Question	Link
Q1. Evaluate each limit by using theorem of limits.	1.3 Q-1
Q2. Evaluate each limit by using algebraic techniques.	1.3 Q-2
Q3. Evaluate the following limits.	1.3 Q-3
Q4. Express each limit in terms of e.	1.3 Q-4

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Overview

In this chapter, we explore the fundamental theorems related to limits, a key concept in calculus. The study of limits is essential for understanding the behavior of functions as they approach specific points. The theorems discussed help simplify the evaluation of limits, especially when direct substitution is not possible. These theorems form the foundation for more advanced topics like continuity, derivatives, and integrals.

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Key Concepts

• Limit of a Function: A limit describes the value a function approaches as its input approaches a certain point.
• Theorems on Limits: These are rules that simplify the evaluation of limits.
  • Sum of Limits: The limit of the sum of two functions is the sum of the limits of the functions.
  • Product of Limits: The limit of the product of two functions is the product of their limits.
  • Quotient of Limits: The limit of the quotient of two functions is the quotient of their limits, given the denominator’s limit is non-zero.
  • Power of a Limit: The limit of a function raised to a power is the limit of the function raised to that power.

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Important Formulas

• Sum of Limits:
  $\lim_{x \to a} \left( f(x) + g(x) \right) = \lim_{x \to a} f(x) + \lim_{x \to a} g(x)$
• Product of Limits:
  $\lim_{x \to a} \left( f(x) \cdot g(x) \right) = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)$
• Quotient of Limits:
  $\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)} \quad \text{(if } \lim_{x \to a} g(x) \neq 0)$
• Power of a Limit:
  $\lim_{x \to a} \left[ f(x) \right]^n = \left[ \lim_{x \to a} f(x) \right]^n$

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Tips and Tricks

• Simplify expressions: When dealing with limits, always try to simplify expressions algebraically before applying the limit.
• Factorization: If the limit involves a rational function, factor both the numerator and the denominator to cancel out common terms before applying the limit.
• Use substitution: If possible, directly substitute the value of $x = a$ in the limit expression to check if the limit can be evaluated easily.

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Summary

In this Exercise, we covered theorems on limits which simplify the evaluation of limits involving sums, products, quotients, and powers. These theorems are indispensable tools in calculus and provide a foundation for understanding more advanced mathematical concepts.

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Reference

By Sir Shazad Sair:

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