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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 \(\mathrm{L}\) ".


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.


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 \(\mathrm{x}=\displaystyle \lim_{\mathrm{t} \rightarrow \mathrm{a}} \mathrm{f}(\mathrm{t})\).


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.


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 \(\displaystyle \lim_{x \rightarrow a} g(x)=\ell\)

Squeeze play theorem (Sandwich theorem):
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 \(a+h\)


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 \(\displaystyle \lim_{x \rightarrow a} \dfrac{\sin f(x)}{f(x)}=1\).


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 \(\displaystyle \lim_{x \rightarrow a}[f(x)]^{\phi(x)}=e^{B \ln A}=A^{B}\)


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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