We need to evaluate the limit:

$\lim_{x \to 0} \frac{\tan(x)}{x}$

Step 1: Use the Standard Limit Result

A well-known trigonometric limit states:

$\lim_{x \to 0} \frac{\sin(x)}{x} = 1$

We can rewrite $\tan(x)$ in terms of sine and cosine:

$\frac{\tan(x)}{x} = \frac{\sin(x)}{x} \cdot \frac{1}{\cos(x)}$

Taking the limit as $x \to 0$:

$\lim_{x \to 0} \frac{\tan(x)}{x} = \lim_{x \to 0} \frac{\sin(x)}{x} \cdot \lim_{x \to 0} \frac{1}{\cos(x)}$

From the standard limit result:

$\lim_{x \to 0} \frac{\sin(x)}{x} = 1$

And since $\cos(0) = 1$, we get:

$\lim_{x \to 0} \frac{1}{\cos(x)} = 1$

Thus, multiplying both limits:

$1 \times 1 = 1$

Final Answer:

$\lim_{x \to 0} \frac{\tan(x)}{x} = 1$

Would you like further explanation or more examples?

Related Questions:

1. What is $\lim_{x \to 0} \frac{\sin(x)}{x}$ and how is it derived?
2. How can L'Hôpital’s Rule be used to evaluate $\lim_{x \to 0} \frac{\tan(x)}{x}$?
3. What are the small-angle approximations for sine and tangent?
4. How does the Taylor series expansion of $\tan(x)$ help in evaluating limits?
5. What is the limit of $\frac{\tan(x) - x}{x^3}$ as $x \to 0$?

Tip:

For small angles, $\sin(x) \approx x$ and $\tan(x) \approx x$. These approximations are useful for quick limit evaluations.