Question:
$ \displaystyle\lim_{x\rightarrow0} \frac{1}{3-2^{\frac{1}{x}}}$ is equal to
$ \displaystyle\lim_{x\rightarrow0} \frac{1}{3-2^{\frac{1}{x}}}$ is equal to
Updated On: Apr 8, 2024
- 0
- $1$
- $\frac{1}{2}$
- $\frac{1}{3}$
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The Correct Option is D
Solution and Explanation
$\displaystyle \lim _{x \rightarrow 0^{-}} \frac{1}{3-2^{1 / x}}$
Put $x =0-h $
$\therefore\displaystyle \lim _{h \rightarrow 0} \frac{1}{3-2^{\frac{1}{0-h}}}$
$ =\displaystyle \lim _{h \rightarrow 0} \frac{1}{3-2^{-\frac{1}{h}}} $
$=\frac{1}{3-2^{-\frac{1}{0}}}=\frac{1}{3-2^{-\infty}} $
$=\frac{1}{3-0}=\frac{1}{3}$
Put $x =0-h $
$\therefore\displaystyle \lim _{h \rightarrow 0} \frac{1}{3-2^{\frac{1}{0-h}}}$
$ =\displaystyle \lim _{h \rightarrow 0} \frac{1}{3-2^{-\frac{1}{h}}} $
$=\frac{1}{3-2^{-\frac{1}{0}}}=\frac{1}{3-2^{-\infty}} $
$=\frac{1}{3-0}=\frac{1}{3}$
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Concepts Used:
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Mathematically, a limit is explained as a value that a function approaches as the input, and it produces some value. Limits are essential in calculus and mathematical analysis and are used to define derivatives, integrals, and continuity.
Limits Formula:
Derivatives of a Function:
A derivative is referred to the instantaneous rate of change of a quantity with response to the other. It helps to look into the moment-by-moment nature of an amount. The derivative of a function is shown in the below-given formula.
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