Question:
The value of $ \displaystyle\lim _{x \rightarrow 0} \frac{\cot 4 x}{\text{cosec} 3 x}$ is equal to
The value of $ \displaystyle\lim _{x \rightarrow 0} \frac{\cot 4 x}{\text{cosec} 3 x}$ is equal to
Updated On: Jun 3, 2024
- $\frac{4}{3}$
- $\frac{3}{4}$
- $\frac{2}{3}$
- $\frac{3}{2}$
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The Correct Option is B
Solution and Explanation
Given, $\displaystyle\lim _{x \rightarrow 0} \frac{\cot 4 x}{\operatorname{cosec} 3 x}$
$=\displaystyle\lim _{x \rightarrow 0}\left(\frac{\sin 3 x}{\tan 4 x}\right)$
$\displaystyle\lim _{x \rightarrow 0} \frac{\frac{\sin 3 x}{3 x} \times 3 x}{\frac{\tan 4 x}{4 x} \times 4 x}$
$=\displaystyle\lim _{x \rightarrow 0} \frac{3 x}{4 x}=\frac{3}{4}$
$=\displaystyle\lim _{x \rightarrow 0}\left(\frac{\sin 3 x}{\tan 4 x}\right)$
$\displaystyle\lim _{x \rightarrow 0} \frac{\frac{\sin 3 x}{3 x} \times 3 x}{\frac{\tan 4 x}{4 x} \times 4 x}$
$=\displaystyle\lim _{x \rightarrow 0} \frac{3 x}{4 x}=\frac{3}{4}$
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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:
Read More: Limits and Derivatives