Limit of a function $f(x)$ is said to exist as $x \rightarrow a$ when,

PYQ-2023-Definite_Integration-Q14

$\quad $ $\lim_{h \to 0^{+}} f(a-h) =\lim_{h \to 0^{+}} f(a+h)=$ some finite value $M$

$\quad \quad$ (Left hand limit) $\quad \quad $ (Right hand limit)

Indeterminant Forms:

$$ \frac{0}{0}, \frac{\infty}{\infty}, 0 \times \infty, \infty-\infty, \infty^{0}, 0^{0} \text {, and } 1^{\infty} \text {. } $$

Standard Limits:

• $$ \lim_{x \to 0} \frac{\sin x}{x} = \lim_{x \to 0} \frac{\tan x}{x} = \lim_{x \to 0} \frac{\tan^{-1}x}{x}=1$$

$$ \lim_{x \to 0} \frac{\sin^{-1}x}{x} = \lim_{x \to 0} \frac{e^{x}-1}{x}=\lim_{x \to 0} \frac{\ell n(1+x)}{x}=1 $$

$$ \lim_{x \to 0} (1+x)^{1 / x}= \lim_{x \to \infty} (1+\frac{1}{x})^{x}=e$$

• $$\lim_{x \to 0} \frac{a^{x}-1}{x}=\log_{e} a, a>0, ~ ~ ~ ~ \lim_{x \to 0} \frac{x^{n}-a^{n}}{x-a}=n a^{n-1}$$

Limits Using Expansion

• $$ a^{x}=1+\frac{x \ln a}{1 !}+\frac{x^{2} \ln ^{2} a}{2 !}+\frac{x^{3} \ln ^{3} a}{3 !}+\ldots \ldots . . a>0$$

• $$ \mathrm{e}^{\mathrm{x}}=1+\frac{\mathrm{x}}{1 !}+\frac{\mathrm{x}^{2}}{2 !}+\frac{\mathrm{x}^{3}}{3 !}+\ldots \ldots$$

• $$\ln (1+x)=x-\frac{x^{2}}{2}+\frac{x^{3}}{3}-\frac{x^{4}}{4}+\ldots \ldots \ldots for -1<x \leq 1$$

• $$\sin x=x-\frac{x^{3}}{3 !}+\frac{x^{5}}{5 !}-\frac{x^{7}}{7 !}+\ldots$$

• $$ \cos x=1-\frac{x^{2}}{2 !}+\frac{x^{4}}{4 !}-\frac{x^{6}}{6 !}+\ldots \ldots$$

• $$ \tan x=x+\frac{x^{3}}{3}+\frac{2 x^{5}}{15}+\ldots \ldots$$

• $$\tan ^{-1} x=x-\frac{x^{3}}{3}+\frac{x^{5}}{5}-\frac{x^{7}}{7}+\ldots$$

• $$\sin ^{-1} x=x+\frac{1^{2}}{3 !} x^{3}+\frac{1^{2} \cdot 3^{2}}{5 !} x^{5}+\frac{1^{2} \cdot 3^{2} \cdot 5^{2}}{7 !} x^{7}+\ldots$$

• For |x|<1,$$(1+x)^{n}=1+n x+\frac{n(n-1)}{1.2} x^{2}+\frac{n(n-1)(n-2)}{1 \cdot 2 \cdot 3} x^{3}+\text{……} \infty$$

• $\quad $ $$(1+x)^n = 1 + nx + \frac{n(n-1)x^2}{2!} + … \text{~ where,~} |x| < 1$$

Limits of form $1^{\infty}, 0^{0}, \infty^{0}$

$\quad $ Also we can use following rules.

• $\lim _{x \rightarrow 0}(1+x)^{1 / x}=e, \quad \lim _{x \rightarrow a}[f(x)]^{g(x)}$, where $f(x) \rightarrow 1 \quad ; \quad g(x) \rightarrow \infty$ as $x \rightarrow a=\lim _{x \rightarrow a}=e^{\lim ^{x \rightarrow a}[f(x)-1] g(x)}$

Sandwich theorem or squeeze play theorem:

• If $f(x) \leq g(x) \leq h(x) \forall x \hspace{1mm} $ & $ \hspace{1mm} \lim_{x \to a} f(x)= \ell = \lim_{x \to a} h(x)$ then $\lim_{x \to a} g(x)=\ell$.

Algebra of limits:

PYQ-2023-Limits-Q3, PYQ-2023-Limits-Q4, PYQ-2023-Definite_Integration-Q6

• $$ \lim_{x \to a}(f+g)(x) = \lim_{x \to a}f(x) + \lim_{x \to a}g(x) $$

• $$ \lim_{x \to a}(f-g)(x) = \lim_{x \to a}f(x) - \lim_{x \to a}g(x) $$

• $$ \lim_{x \to a}(c f)(x) = c \lim_{x \to a}f(x) \text{where, c is a constant} $$

• $$ \lim_{x \to a}(fg)(x) = [\lim_{x \to a}f(x)] [\lim_{x \to a}g(x)] $$

• $$ \lim_{x \to a}\left(\frac{f}{g}\right)(x) = [\frac{\lim_{x \to a}f(x)} {\lim_{x \to a}g(x)}] $$

• $$ \lim_{x \to a}(f(x))^n = (\lim_{x \to a}f(x))^n$$

• If $$ f(x) \leq g(x) \text{~then,~} \lim_{x \to a} f(x) \leq \lim_{x \to a} g(x) $$

Standard results:

PYQ-2023-Limits-Q2, PYQ-2023-Limits-Q5

• $$ \lim_{x \to a} \frac{x^n - a^n}{x-a} = n a^{n-1}$$

• $$ \lim_{x \to a} \frac{x^n - a^n}{x-a} = n a^{n-1}$$

• $$ \lim_{x \to a} \frac{x^n - a^n}{x-a} = n a^{n-1}$$

• $$ \lim_{x \to a} \frac{x^n - a^n}{x^m-a^m} = \frac{n}{m} a^{n-m}$$

$\qquad$

• If $\lim_{x \to a} f(x)=1$ and $\lim_{x \to a} g(x)= \infty$ then,

$$ \lim_{x \to a} [f(x)]^{g(x)}= e^{\lim_{x \to a}g(x)[f(x)-1]} $$

• $$\lim_{x\to0}(1+ax)^{\frac{1}{x}}=e^a=\lim_{x\to\infty}(1+\frac{a}{x})^{x}$$

L’Hospital’s rule:

PYQ-2023-Limits-Q1, PYQ-2023-Limits-Q3, PYQ-2023-Definite_Integration-Q14

$\quad $ For functions f and g which are differentiable:

$\quad$ If $\lim_{x \to c} f(x)= \lim_{x \to c}g(x) = 0 ~ \text{or } ~ \pm \infty$ $\lim_{x \to c}\frac{f^{\prime}}{g^{\prime}}$ has a finite value then $\lim_{x \to c}\frac{f^{\prime}}{g^{\prime}}=\lim_{x \to c}\frac{f}{g} $