Question:
$\displaystyle\lim_{x \to 0} \frac{xe^x - \sin \, x}{x}$ is equal to
$\displaystyle\lim_{x \to 0} \frac{xe^x - \sin \, x}{x}$ is equal to
Updated On: Apr 18, 2024
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The Correct Option is C
Solution and Explanation
$\displaystyle\lim _{x \rightarrow 0} \frac{x e^{x}-\sin\, x}{x} \,\,\left(\frac{0}{0}\text{form} \right)$
$=\displaystyle\lim _{x \rightarrow 0} \frac{x e^{x}+e^{x}-\cos\, x}{1} \,\,$ [using L' Hospital's rule]
$=0+1-1=0$
$=\displaystyle\lim _{x \rightarrow 0} \frac{x e^{x}+e^{x}-\cos\, x}{1} \,\,$ [using L' Hospital's rule]
$=0+1-1=0$
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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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