JEE Main & Advanced Mathematics Functions Fundamental Theorems on Limits

Fundamental Theorems on Limits

Category : JEE Main & Advanced

The following theorems are very useful for evaluation of limits if $\underset{x\to 0}{\mathop{\lim }}\,f(x)=l$ and $\underset{x\to 0}{\mathop{\lim }}\,g(x)=m$ ($l$ and $m$ are real numbers) then

(1) $\underset{x\to a}{\mathop{\lim }}\,(f(x)+g(x))=l+m\,$                              (Sum rule)

(2) $\underset{x\to a}{\mathop{\lim }}\,(f(x)-g(x))=l-m$                  (Difference rule)

(3) $\underset{x\to a}{\mathop{\lim }}\,(f(x).g(x))=l.m$                   (Product rule)

(4) $\underset{x\to a}{\mathop{\lim }}\,k\,\,f(x)=k.l$                                      (Constant multiple rule)

(5) $\underset{x\to a}{\mathop{\lim }}\,\,\frac{f(x)}{g(x)}=\frac{l}{m},m\ne 0$                             (Quotient rule)

(6) If $\underset{x\to a}{\mathop{\lim }}\,f(x)=+\infty$ or $-\infty$, then $\underset{x\to a}{\mathop{\lim }}\,\,\frac{1}{f(x)}=0$

(7) $\underset{x\to a}{\mathop{\lim }}\,\log \{f(x)\}=\log \,\{\underset{x\to a}{\mathop{\lim }}\,f(x)\}$

(8) If $f(x)\le g(x)$ for all $x,$ then $\underset{x\to a}{\mathop{\lim }}\,f(x)\le \underset{x\to a}{\mathop{\lim }}\,g(x)$

(9) $\underset{x\to a}{\mathop{\lim }}\,{{[f(x)]}^{g(x)}}={{\{\underset{x\to a}{\mathop{\lim }}\,f(x)\}}^{\underset{x\to a}{\mathop{\lim }}\,g(x)}}$

(10) If $p$ and $q$ are integers, then $\underset{x\to a}{\mathop{\lim }}\,{{(f(x))}^{p/q}}={{l}^{p/q}},$ provided ${{(l)}^{p/q}}$ is a real number.

(11) If $\underset{x\to a}{\mathop{\lim }}\,f(g(x))=f(\underset{x\to a}{\mathop{\lim }}\,g(x))=f(m)$ provided $'f'$ is continuous at $g(x)=m.\,\,e.g.$$\underset{x\to a}{\mathop{\lim }}\,\ln [f(x)]=\ln (l),$only if $l>0.$

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