Question:
$\displaystyle \lim_{x\to0} \frac{sin\left(\pi\,cos^{2} x\right)}{x^{2}}$ equals
$\displaystyle \lim_{x\to0} \frac{sin\left(\pi\,cos^{2} x\right)}{x^{2}}$ equals
Updated On: Jul 14, 2022
- $-\pi$
- $1$
- $-1$
- $\pi$
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The Correct Option is D
Solution and Explanation
Consider
$\displaystyle \lim_{x\to0} \frac{sin\left(\pi\,cos^{2} x\right)}{x^{2}}$
$= \displaystyle \lim_{x\to0} \frac{sin\left(\pi-\pi\,sin^{2} x\right)}{x^{2}} $
$\left[\because sin \left(\pi - \theta\right) = sin \,\theta\right]$
$= \displaystyle \lim _{x\to 0} \frac{sin\left(\pi \,sin^{2} x\right)}{\pi \,sin^{2}x}\times\frac{\left(\pi \,sin^{2} x\right)}{x^{2}} = \pi$
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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.
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