Question

$$ f(x)=\frac{\sin ^{-1} 2 x}{x} ; \lim _{x \rightarrow 0} f(x) $$

   $$
f(x)=\frac{\sin ^{-1} 2 x}{x} ; \lim _{x \rightarrow 0} f(x)
$$

Calculus Early Transcendentals

Calculus Early Transcendentals

Howard Anton, Irl C.… 10th Edition

Chapter 1, Problem 12

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Solved by Expert Aman Gupta

02/22/2023

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Step 1: We are given the function $f(x)=\frac{\sin ^{-1} 2 x}{x}$ and we need to find $\lim _{x \rightarrow 0} f(x)$. Show more…

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$$ f(x)=\frac{\sin ^{-1} 2 x}{x} ; \lim _{x \rightarrow 0} f(x) $$

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Key Concepts

L'Hôpital's Rule

L'Hôpital's Rule is a method used to evaluate limits that result in indeterminate forms like 0/0 or ?/?. It involves differentiating the numerator and denominator separately and then taking the limit of the resulting expression, providing a systematic way to resolve such indeterminate cases.

Small-Angle Approximation

The small-angle approximation is a technique used when dealing with trigonometric functions for very small angles, where functions like arcsin(x) can be approximated by their argument. This approximation simplifies the evaluation of limits as it leverages the fact that the function behaves nearly linearly close to zero.

Limit Evaluation

Limits are a fundamental concept in calculus that describe the behavior of a function as its argument approaches a specific value. In problems involving limits, especially at points where direct substitution leads to indeterminate forms, various techniques are applied to analyze the function's behavior and compute the limit accurately.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as arcsin, are the inverse operations of the trigonometric functions. Understanding the properties and behavior of these functions, especially near zero where they can be approximated linearly, is essential in evaluating limits and solving related problems in calculus.

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