Question

Find the limit, if it exists. $$ \lim _{x \rightarrow 0} \frac{\sin x}{x-\tan x} $$

   Find the limit, if it exists.
$$
\lim _{x \rightarrow 0} \frac{\sin x}{x-\tan x}
$$

Calculus

Earl W. Swokowski 5th Edition

Chapter 10, Problem 10

Instant Answer

Solved by Expert Gregory Higby

03/19/2022

Step 1

We get $\frac{\sin(0)}{0-\tan(0)}$, which simplifies to $\frac{0}{0}$. This is an indeterminate form, so we can apply L'Hopital's rule. Show more…

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Find the limit, if it exists. $$ \lim _{x \rightarrow 0} \frac{\sin x}{x-\tan x} $$

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Key Concepts

Indeterminate Forms

An indeterminate form occurs when the limit of a function results in an expression that does not have a clear or unique value, such as 0/0 or ?/?. Recognizing these forms is critical because they indicate that direct substitution is insufficient, necessitating additional techniques to evaluate the limit.

Taylor Series Expansion

The Taylor series is a method used to approximate functions around a point by expressing them as an infinite sum of derived terms based on derivatives at that point. In limit problems, particularly those yielding an indeterminate form, the series expansion can simplify complex functions into polynomial forms, making it easier to evaluate the limit.

Limit

In calculus, a limit describes the behavior of a function as its input approaches a certain value. It is an essential concept for defining continuity, derivatives, and integrals, particularly when the function does not have an explicit value at a point or behaves in an indeterminate way.

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