https://file.aiquickdraw.com/mathbot/file/17264605405648gpv1ayd.jpg

The expression shown in the image is:

\[
\lim_{{x \to 0}} \frac{\tan x}{x}
\]

To solve this limit:

1. **Direct Substitution**: Substituting \(x = 0\) directly into the expression \(\frac{\tan x}{x}\) results in an indeterminate form \(\frac{0}{0}\).

2. **Using Limit Identity**: There's a well-known trigonometric limit identity:

\[
\lim_{{x \to 0}} \frac{\tan x}{x} = 1
\]

Thus, the value of the limit is:

\[
\lim_{{x \to 0}} \frac{\tan x}{x} = 1
\]

Would you like further details or explanations about how this result is derived?

Here are 5 related questions to explore:

1. How do you compute limits involving other trigonometric functions?
2. Why is L'Hôpital's Rule useful for indeterminate forms like \(\frac{0}{0}\)?
3. What is the difference between the limit of \(\frac{\sin x}{x}\) and \(\frac{\tan x}{x}\) as \(x \to 0\)?
4. How can you evaluate limits of functions using series expansion?
5. What other common limit identities should be memorized for calculus?

**Tip:** When faced with indeterminate forms like \(\frac{0}{0}\), try using limit identities or L'Hôpital’s Rule for a solution.

Learn how to solve the limit \(\lim_{{x \to 0}} \frac{\tan x}{x}\), which results in 1. This solution covers important trigonometric limit identities and is suitable for students studying advanced high school mathematics or early calculus.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation