Question:
The value of $ \displaystyle\lim_{x\to\infty} \left(\frac{x+5}{x+2}\right)^{x} $ is equal to
The value of $ \displaystyle\lim_{x\to\infty} \left(\frac{x+5}{x+2}\right)^{x} $ is equal to
Updated On: Jun 23, 2024
- $ e^{-3} $
- $ e $
- $ e^{3} $
- $ e^{2} $
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The Correct Option is C
Solution and Explanation
We have,$\lim\limits_{x\to\infty}\left(\frac{x+5}{x+2}\right)^{x} $
$=\lim\limits_{x\to\infty}\left(1+\frac{3}{x+2}\right)^{x} (1^{\infty}$ form) $=e^{\lim\limits_{x\to\infty} \frac{3x}{2}} = e^{\lim\limits_{x\to\infty} \frac{3}{\left(1+2/ x\right)}} = e^{3}$
$=\lim\limits_{x\to\infty}\left(1+\frac{3}{x+2}\right)^{x} (1^{\infty}$ form) $=e^{\lim\limits_{x\to\infty} \frac{3x}{2}} = e^{\lim\limits_{x\to\infty} \frac{3}{\left(1+2/ x\right)}} = e^{3}$
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Concepts Used:
Limits And Derivatives
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.
Properties of Derivatives:
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