Limits
1.0 Introduction
1.1 Basic Method of Evaluation of Limits:
1.2 Questions
1.3 Formal definition of Limit:
1.4 Evaluation of Limits by Direct Substitution Method:
1.5 Neighbourhood Concept:
2.0 Definition of Limit - In a different form:
2.1 One - Sided Limits:
2.2 Left hand Limit of a function:(LHL)
2.3 Right hand Limit of a function:(RHL)
3.0 Conditions for existence of Limit
4.0 Some Standard Limits
5.0 Algebra of limits
6.0 Some Standard Methods of Evaluation of Limits:
7.0 Indeterminate Forms:
7.1 Limits of the form ${1^\infty }$
7.2 Limits of the form ${0^0}$
7.3 Limits of the form ${\infty^0}$
7.4 Limit of a function as $x \to \infty $ :
8.0 Sandwich Theorem / Squeeze Play Theorem:
9.0 L'Hospital's Rule for evaluation of limits:
7.1 Limits of the form ${1^\infty }$
1.2 Questions
1.3 Formal definition of Limit:
1.4 Evaluation of Limits by Direct Substitution Method:
1.5 Neighbourhood Concept:
2.2 Left hand Limit of a function:(LHL)
2.3 Right hand Limit of a function:(RHL)
7.2 Limits of the form ${0^0}$
7.3 Limits of the form ${\infty^0}$
7.4 Limit of a function as $x \to \infty $ :
Let $f(x)$ and $g(x)$ be functions of $x$.
Let $\mathop {\lim }\limits_{x \to a} f(x) = 1$ and $\mathop {\lim }\limits_{x \to a} g(x) = \infty $.
$ \Rightarrow $ $\mathop {\lim }\limits_{x \to a} f{(x)^{g(x)}}$ will be in the form of ${1^\infty }$.
Then
$\mathop {\lim }\limits_{x \to a} f{(x)^{g(x)}} = {e^{\mathop {\lim }\limits_{x \to a} \left( {f(x) - 1} \right)g(x)}}$
Question 18.
Evaluate $\mathop {\lim }\limits_{x \to \infty } {\left( {1 + \frac{2}{x}} \right)^x}$.
Solution:
$\mathop {\lim }\limits_{x \to \infty } {\left( {1 + \frac{2}{x}} \right)^x}$ is in the form of ${1^\infty }$.
$ \Rightarrow $ $\mathop {\lim }\limits_{x \to \infty } {\left( {1 + \frac{2}{x}} \right)^x} = {e^{\mathop {\lim }\limits_{x \to \infty } \left( {1 + \frac{2}{x} - 1} \right)x}} = {e^2}$
