Introduction
Limit and continuity is the introduction of calculus. Calculus is
that branch of Mathematics which mainly deals with the study of change in the
value of a function as the point in the domain changes.
Definition:
Suppose f is a real function on a subset of the real numbers and
let c be a point in the domain of f. Then f is continuous at c
if \[\lim_{x\rightarrow c}f(x)=f(c)\]
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ALGEBRA OF LIMITS |
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\[Let \; f(x)\; and\; g(x)\; be\; two\;
functions \; such\; that\; \lim_{x\rightarrow 0}\; f(x)\; and\;
\lim_{x\rightarrow 0}\; g(x) \; exists\]\[1)\; \lim_{x\rightarrow a}\left [
f(x)+g(x) \right ]=\lim_{x\rightarrow a}\: f(x)+\lim_{x\rightarrow a}\: g(x)\]\[2)\;
\lim_{x\rightarrow a}\left [ f(x)-g(x) \right ]=\lim_{x\rightarrow a}\:
f(x)-\lim_{x\rightarrow a}\: g(x)\]\[3)\; \lim_{x\rightarrow a}\left [
f(x)\times g(x) \right ]=\lim_{x\rightarrow a}\: f(x)\times
\lim_{x\rightarrow a}\: g(x)\]\[4)\; \lim_{x\rightarrow a}
\frac{f(x)}{g(x)}=\frac{\lim_{x\to a}f(x)}{\lim_{x\to a}g(x)}\]Limit of the
constant function remain same constant. \[\lim_{x\rightarrow c}24=24\] |
Undefined terms: \[\frac{0}{0},\; \; \frac{\infty }{\infty },\; \; 0^{0},\; \; \infty ^{\infty },\; \; \: \: all\: are\: called\: undefined\: form\]If we put the limit and we got undefined form then at that point limit cannot be calculated. In this case we first simplify the functions by applying some formulas and then again find the limit.
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IMPORTANT FORMULAS USED IN LIMITS |
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\[\lim_{x\to
a}\frac{x^{n}-a^{n}}{x-a}=na^{n-1}\]\[\lim_{x\to 0
}\frac{e^{x}-1}{x}=1\]\[\lim_{\lambda x\to0 }\frac{e^{\lambda x-1}}{\lambda
x}=\lambda \: \: \: where\: \lambda \neq 0\]\[\lim_{x\to0
}\frac{Log(1+x)}{x}= 1,\: \lim_{x\to e }Log_{e}\: x=1\]\[\lim_{x\to0
}\frac{Log_{e}(1-x)}{x}= -1,\] |
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LIMITS OF TRIGONOMETRIC FUNCTIONS |
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\[\lim_{x\to 0}sinx=0, \: \: \: \:\; \;
\lim_{x\to 0}cosx=1\]\[\lim_{x\to 0}\: \frac{sinx}{x}=1,\: \: \: \lim_{x\to
0}\frac{x}{sinx}=1, \\ \lim_{x\to 0}\:
\frac{tanx}{x}=1,\lim_{x\to0}\frac{x}{tanx}=1\]\[\lim_{x\to0}\frac{sin^{-1}x}{x}=1,\;\:
\lim_{x\to0}\frac{x}{sin^{-1}x}=1\]\[\lim_{x\to0}\frac{tan^{-1}x}{x}=1,\;\:
\lim_{x\to0}\frac{x}{tan^{-1}x}=1\]\[\lim_{x\to0}\frac{sinx^{o}}{x}=\frac{\pi
}{180^{o}}\]\[\lim_{x\to a}\frac{sin(x-a))}{x-a}=1=\lim_{x\to
a}\frac{tan(x-a))}{x-a}\]*********************************************
*******\[\lim_{x\to\infty }\frac{sinx}{x}=\: 0\: =\lim_{x\to\infty }\frac{cosx}{x}\]\[\lim_{x\to\infty
}\frac{sin\frac{1}{x}}{\frac{1}{x}}=\: 0\: =\lim_{x\to\infty
}\frac{cosx}{x}\] |
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Left Hand Limit and Right Hand Limit
If at any point c given function have two different values then at that point (c) we calculate LHL and RHL. and the limit exist if
Left Hand Limit = Right Hand Limit = f(c)\[LHL \; is\; denoted\; by\; \lim_{x\rightarrow c^{-}}f(x)\]\[RHL \; is\; denoted\; by\; \lim_{x\rightarrow c^{+}}f(x)\]\[When \; limit\; exists\; then\; \lim_{x\rightarrow c^{-}}f(x)=\lim_{x\rightarrow c^{+}}f(x)=f(c)\]\[For\; a\; function\; f(x)\; the\; limits\; \lim_{x\to a^{-}}f(x)\; and\; \lim_{x\to a^{+}}f(x),\; may\; or \; may \; not\; exists\]\[If both\; limits\; \lim_{x\to a^{-}}f(x)\; and\; \lim_{x\to a^{+}}f(x)\; exists, they \; may \; or \; may\; not\; be \; equal\]\[If\;\; \lim_{x\to a^{-}}f(x)\; \neq \; \lim_{x\to a^{+}}f(x)\; ,then\; \lim_{x\to a}\; does \; not\; exist\]\[If\;\; \lim_{x\to a^{-}}f(x)\; = \; \lim_{x\to a^{+}}f(x)\; ,then\; \lim_{x\to a}\; \; exist\; and\; is\; equal\; to\; the\; common\; limit\]***********************************************
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CONTINUITY OF A
FUNCTION: |
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A function is said to be continuous if its limit exist
or if \[\lim_{x \to c}f(x)=f(c)\; \; or\; \;if \]\[Left\; \;
Hand\; \; Limit = Right\; \; Hand\; \; Limit = f(c)\] |
EVERYWHERE CONTINUOUS FUNCTION
Constant Functions
Exponential Functions
Polynomial Functions
Modulus Functions
Rational Functions
Trigonometric Functions
Logarithmic Functions
EXPONENTIAL LIMITS\[Here\; we\; use\; the\; series:\:\: \: e^{x}=1+x+\frac{x^{2}}{2!}+\frac{x^{3}}{3!}+.......\infty\]\[\lim_{x\to \infty }\frac{1}{x}=0\; \; \; \; \; \; \lim_{x\to \infty }\frac{1}{x^{k}}=0\]\[\lim_{x\to 0 }(1+x)^{\frac{1}{x}}=e\]\[\lim_{x\to0 }(1+\lambda x)^{\frac{1}{x}}=e^{\lambda }\]\[\lim_{x\to\infty }(1+\frac{1}{x})^{x}=e,\: \: \lim_{x\to\infty }(1+\frac{\lambda }{x})^{x}=e^{\lambda }\]
LOGARITHMIC LIMITS \[Here\; we\; use\; the \; series:\: \: Log(1+x)=x-\frac{x^{2}}{2}+\frac{x^{3}}{3}-......\infty\]\[\lim_{x\to0 }\frac{Log_{a}(1+x)}{x}= Log_{a}\: e\]\[\lim_{x\to 0 }\frac{a^{x}-1}{x}=log\: a\: \: where\: \: a> 0\]
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DERIVATIVE BY FIRST PRINCIPAL |
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\[f^{'}(x)=\lim_{ h \to0}\frac{f(x+h)-f(x)}{h}\; (OR)\;
f^{'}(x)=\lim_{x\to a}\frac{f(x)-f(a)}{x-a}\]
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DIFFERENTIATION OF SOME IMPORTANT
FUNCTIONS |
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\[\frac{d}{dx}(x)=1\] \[\frac{d}{dy}(x^{2})=2x,\;
\; \; \frac{d}{dx}(x^{3})=3x^{2},\; \; \; \frac{d}{dx}(x^{4})=4x^{3}\] \[\frac{d}{dx}x^{n}=nx^{n-1}\] \[\frac{d}{dx}(ax+b)=a\] \[\frac{d}{dx}\left
(ax+b \right )^{3}=3(ax+b)^{3-1}\frac{d}{dx}(ax+b)=3a(ax+b)^{2}\] \[\frac{d(constant)}{dx}=\frac{d(C)}{dx}=0\] \[\frac{d}{dx}e^{x}=e^{x}\] \[\frac{d}{dx}logx=\frac{1}{x}\] \[\frac{d}{dx}\left
( \frac{1}{x} \right )=\frac{-1}{x^{2}}\] \[\frac{d}{dx}\left
( \frac{1}{x^{2}} \right )=\frac{-2}{x^{3}}\] \[\frac{d}{dx}\left
( \frac{1}{x^{3}} \right )=\frac{-3}{x^{4}}\] \[\frac{d}{dx}\left
( \frac{1}{x^{n}} \right )=\frac{-n}{x^{n+1}}\] |
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PRODUCT RULE OF DIFFERENTIATION |
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\[\left ( uv \right )^{'}=u^{'}v+uv^{'}\]
or \[\frac{d(uv)}{dx}=\left (\frac{du}{dx} \right )v+u\left ( \frac{dv}{dx}
\right )\] |
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QUOTIENT RULE OF DIFFERENTIATION |
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\[\left ( \frac{u}{v} \right
)^{'}=\frac{u'v-uv'}{v^{2}}-\] or \[\frac{d}{dx}\left ( \frac{u}{v}
\right )=\frac{\left (\frac{d}{dx}u \right )v-u\left ( \frac{d}{dx}v \right
)}{v^{2}}\] |
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DIFFERENTIATION OF SOME TRIGONOMETRIC
FUNCTIONS |
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\[\frac{d}{dx}sinx=cosx\] \[\frac{d}{dx}cosx=-sinx\] \[\frac{d}{dx}tanx=sec^{2}x\] \[\frac{d}{dx}cotx=-cosec^{2}x\] \[\frac{d}{dx}secx=secx\;
tanx\] \[\frac{d}{dx}cosecx=-cosecx\;
cotx\] |
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