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

$\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)

Algebra of limits:

• $ \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) $

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$

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

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

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

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$

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 $\lim_{x \to a} g(x)=\ell$.

L’Hospital’s rule:

$\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} $