Limit - Notes, Concept and All Important Formula

LIMIT

1. DEFINITION :

Let $\mathrm{f}(\mathrm{x})$ be defined on an open interval about 'a' except possibly at 'a' itself. If $\mathrm{f}(\mathrm{x})$ gets arbitrarily close to $\mathrm{L}$ (a finite number) for all $\mathrm{x}$ sufficiently close to 'a' we say that $\mathrm{f}(\mathrm{x})$ approaches the limit $\mathrm{L}$ as $\mathrm{x}$ approaches 'a' and we write $\displaystyle \lim_{x \rightarrow a} f(x)=L$ and say "the limit of $f(x)$, as $\mathrm{x}$ approaches a, equals $\mathrm{L}$ ".

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All Chapter Notes, Concept and Important Formula

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2. LEFT HAND LIMIT & RIGHT HAND LIMIT OF A FUNCTION:

Left hand limit $(\mathrm{LHL})=\displaystyle \lim_{\mathrm{x} \rightarrow \mathrm{a}^{-}} \mathrm{f}(\mathrm{x})=\displaystyle \lim_{\mathrm{h} \rightarrow 0} \mathrm{f}(\mathrm{a}-\mathrm{h}), \mathrm{h}>0$

Right hand limit $(\mathrm{RHL})=\displaystyle \lim_{\mathrm{x} \rightarrow \mathrm{a}^{+}} \mathrm{f}(\mathrm{x})=\displaystyle \lim_{\mathrm{h} \rightarrow 0} \mathrm{f}(\mathrm{a}+\mathrm{h}), \mathrm{h}>0$

Limit of a function $\mathbf{f}(\mathbf{x})$ is said to exist as $\mathbf{x} \rightarrow \mathbf{a}$ when $\mathbf {\displaystyle \lim_{x \rightarrow a^{-}} f(x)=\displaystyle \lim_{x \rightarrow a^{+}} f(x)=}$ Finite and fixed quantity.

Important note :

In $\mathbf{\displaystyle \lim_{\mathbf{x} \rightarrow \mathbf{a}} \mathbf{f}(\mathbf{x}), \mathbf{x} \rightarrow}$ a necessarily implies $\mathbf{x} \neq \mathbf{a}$. That is while evaluating limit at $\mathrm{x}=\mathrm{a}$, we are not concerned with the value of the function at $x=a$. In fact the function may or may not be defined at $x=a$

Also it is necessary to note that if $\mathrm{f}(\mathrm{x})$ is defined only on one side of  $\mathbf{'x}=\mathbf{a'}$, one sided limits are good enough to establish the existence of limits, & if $\mathrm{f}(\mathrm{x})$ is defined on either side of 'a' both sided limits are to be considered.

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3. FUNDAMENTAL THEOREMS ON LIMITS :

Let $\displaystyle \lim_{x \rightarrow a} f(x)=l$ & $\displaystyle \lim _{x \rightarrow a} g(x)=m$. If $l$ & $m$ exists finitely then :

(a) Sum rule: $\displaystyle \lim_{x \rightarrow a}[f(x)+g(x)]=l+m$

(b) Difference rule : $\displaystyle \lim_{x \rightarrow a}[f(x)-g(x)]=l-m$

(c) Product rule : $\displaystyle \lim_{x \rightarrow a} f(x) \cdot g(x)=l . m$

(d) Quotient rule :$\displaystyle \lim_{\mathrm{x} \rightarrow \mathrm{a}} \dfrac{\mathrm{f}(\mathrm{x})}{\mathrm{g}(\mathrm{x})}=\dfrac{l}{\mathrm{~m}}$, provided $\mathrm{m} \neq 0$

(e) Constant multiple rule :$\displaystyle \lim_{x \rightarrow a} \operatorname{kf}(x)=k\displaystyle \lim_{x \rightarrow a} f(x) ;$ where $k$ is constant.

(f) Power rule: $\displaystyle \lim_{x \rightarrow a}[f(x)]^{g(x)}=l^{\mathrm{m}}$, provided $l>0$

(g) $\displaystyle \lim_{x \rightarrow a} f[g(x)]=f\left(\displaystyle \lim_{x \rightarrow a} g(x)\right)=f(m) ;$ provided $f(x)$ is continuous at $x=m$

For example : $\displaystyle \lim_{x \rightarrow a} \ell n(f(x))=\ell n\left[\displaystyle\lim_{x \rightarrow a}f(x)\right] ;$ provided $\ell n x$ is defined at $\mathrm{x}=\displaystyle \lim_{\mathrm{t} \rightarrow \mathrm{a}} \mathrm{f}(\mathrm{t})$.

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4. INDETERMINATE FORMS :

$\dfrac{0}{0}, \dfrac{\infty}{\infty}, \infty-\infty, 0 \times \infty, 1^{\infty}, 0^{0}, \infty^{0}$

Note:
We cannot plot $\infty$ on the paper. Infinity $(\infty)$ is a symbol & not a number. It does not obey the laws of elementary algebra.

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5. GENERAL METHODS TO BE USED TO EVALUATE LIMITS:

(a) Factorization :

Important factors:
(i) $x^{n}-a^{n}$$=(x-a)\left(x^{n-1}+a x^{n-2}+\ldots \ldots \right.$$\left.\ldots . .+a^{n-1}\right), n \in N$
(ii) $x^{n}+a^{n}$$=(x+a)\left(x^{n-1}-a x^{n-2}+\ldots \ldots\right.$$\left. \ldots . .+a^{n-1}\right), n$ is an odd natural number.

Note: $\displaystyle \lim_{x \rightarrow a} \dfrac{x^{n}-a^{n}}{x-a}=n a^{n-1}$

(b) Rationalization or double rationalization:

In this method we rationalise the factor containing the square root and simplify.

(c) Limit when $x \rightarrow \infty$ :

(i) Divide by greatest power of $\mathrm{x}$ in numerator and denominator.
(ii) Put $x=1 / y$ and apply $y \rightarrow 0$

(d) Squeeze play theorem (Sandwich theorem):

If $f(x) \leq g(x) \leq h(x) ; \,\, \forall \,\, x$ &  $\displaystyle \lim_{x \rightarrow a} f(x)=\ell$$=\displaystyle \lim_{x \rightarrow a} h(x)$ then $\displaystyle \lim_{x \rightarrow a} g(x)=\ell$

For example : $\displaystyle \lim_{x \rightarrow 0} x^{2} \sin \dfrac{1}{x}=0$, as illustrated by the graph given.

(e) Using substitution $\displaystyle \lim_{x \rightarrow a} f(x)=$$\displaystyle \lim_{h \rightarrow 0} f(a-h)$ or $\displaystyle \lim_{h \rightarrow 0} f(a+h)$ i.e. by substituting $x$ by $a-h$ or $a+h$

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6. LIMIT OF TRIGONOMETRIC FUNCTIONS:

$\displaystyle \lim_{x \rightarrow 0} \dfrac{\sin x}{x}=1$$=\displaystyle \lim_{x \rightarrow 0} \dfrac{\tan x}{x}$$=\displaystyle \lim_{x \rightarrow 0} \dfrac{\tan ^{-1} x}{x}$$=\displaystyle \lim_{x \rightarrow 0} \dfrac{\sin ^{-1} x}{x}$ [where $\mathrm{x}$ is measured in radians]
Further if $\displaystyle \lim_{x \rightarrow a} f(x)=0$, then $\displaystyle \lim_{x \rightarrow a} \dfrac{\sin f(x)}{f(x)}=1$.

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7. LIMIT OF EXPONENTIAL FUNCTIONS :

(a) $\displaystyle \lim_{x \rightarrow 0} \dfrac{a^{x}-1}{x}=\ln a (a>0)$. In particular $\displaystyle \lim_{x \rightarrow 0} \dfrac{e^{x}-1}{x}=1$.
In general if $\displaystyle \lim_{x \rightarrow a} f(x)=0$,then $\displaystyle \lim_{x \rightarrow a} \dfrac{a^{f(x)}-1}{f(x)}=\ell$ na, $a>0$.

(b) $\displaystyle \lim_{x \rightarrow 0} \dfrac{\ln (1+x)}{x}=1$

(c) $\displaystyle \lim_{x \rightarrow 0}(1+x)^{1 / x}=e=\displaystyle \lim_{x \rightarrow \infty}\left(1+\dfrac{1}{x}\right)^{x}$

(Note : The base and exponent depends on the same variable.)
In general, if $\displaystyle \lim_{x \rightarrow a} f(x)=0$, then $\displaystyle \lim_{x \rightarrow a}(1+f(x))^{1 / f(x)}=e$

(d) If $\displaystyle \lim_{\mathrm{x} \rightarrow \mathrm{a}} \mathrm{f}(\mathrm{x})=1$ and $\displaystyle \lim_{\mathrm{x} \rightarrow \mathrm{a}} \phi(\mathrm{x})=\infty$,
then $\displaystyle \lim_{x \rightarrow a}[f(x)] \phi(x)=e^{k}$ where $k=\displaystyle \lim_{x \rightarrow a} \phi(x)[f(x)-1]$

(e) If $\displaystyle \lim_{x \rightarrow a} f(x)=A>0 \& \displaystyle \lim_{x \rightarrow a} \phi(x)=B$ (a finite quantity),
then $\displaystyle \lim_{x \rightarrow a}[f(x)]^{\phi(x)}=e^{B \ln A}=A^{B}$

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8. LIMIT USING SERIES EXPANSION :

Binomial expansion, exponential & logarithmic expansion, expansion of $\sin x, \cos x$, tanx should be remembered by heart which are given below:

(a) $\mathrm{a}^{\mathrm{x}}=1$$+\dfrac{\mathrm{x} \ell \mathrm{na}}{1 !}$$+\dfrac{\mathrm{x}^{2} \ln ^{2} \mathrm{a}}{2 !}$$+\dfrac{\mathrm{x}^{3} \ln ^{3} \mathrm{a}}{3 !}$$+\ldots, \mathrm{a}>0$

(b) $e^{x}=1$$+\dfrac{x}{1 !}$$+\dfrac{x^{2}}{2 !}$$+\dfrac{x^{3}}{3 !}$$+\ldots, x \in R$

(c) $\ln (1$$+\mathrm{x})=\mathrm{x}-\dfrac{\mathrm{x}^{2}}{2}$$+\dfrac{\mathrm{x}^{3}}{3}-\dfrac{\mathrm{x}^{4}}{4}$$+\ldots$, for $-1<\mathrm{x} \leq 1$

(d) $\sin x=x-\dfrac{x^{3}}{3 !}$$+\dfrac{x^{5}}{5 !}-\dfrac{x^{7}}{7 !}$$+\ldots, x \in R$

(e) $\cos x=1-\dfrac{x^{2}}{2 !}$$+\dfrac{x^{4}}{4 !}-\dfrac{x^{6}}{6 !}$$+\ldots, x \in R$

(f) $\tan x=x$$+\dfrac{x^{3}}{3}$$+\dfrac{2 x^{5}}{15}$$+\ldots,|x|<\dfrac{\pi}{2}$

(g) $\tan ^{-1} \mathrm{x}=\mathrm{x}-\dfrac{\mathrm{x}^{3}}{3}$$+\dfrac{\mathrm{x}^{5}}{5}-\dfrac{\mathrm{x}^{7}}{7}$$+\ldots, \mathrm{x} \in \mathrm{R}$

(h) $\sin ^{-1} x=x$$+\dfrac{1^{2}}{3 !} x^{3}$$+\dfrac{1^{2} \cdot 3^{2}}{5 !} x^{5}$$+\dfrac{1^{2} \cdot 3^{2} \cdot 5^{2}}{7 !} x^{7}$$+\ldots, x \in[-1,1]$

(i) $(1$$+x)^{n}=1$$+n x$$+\dfrac{n(n-1)}{2 !} x^{2}$$+\ldots, n \in R, x \in(-1,1)$.

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