`lim_(x to 0)(tanx)/x` is :

Options

A. 1
B. 0
C. `infty`
D. None of these

Answer

To solve the limit \( \lim_{x \to 0} \frac{\tan x}{x} \), we can follow these steps:

### Step 1: Identify the form of the limit
When we substitute \( x = 0 \) into the expression \( \frac{\tan x}{x} \), we get:
\[ \tan(0) = 0 \quad \text{and} \quad 0 \Rightarrow \frac{0}{0} \text{ (indeterminate form)} \]
This means we can apply L'Hôpital's Rule.

### Step 2: Apply L'Hôpital's Rule
L'Hôpital's Rule states that if we have an indeterminate form \( \frac{0}{0} \), we can differentiate the numerator and the denominator separately.

Differentiate the numerator \( \tan x \):
\[ \frac{d}{dx}(\tan x) = \sec^2 x \]

Differentiate the denominator \( x \):
\[ \frac{d}{dx}(x) = 1 \]

Now we can rewrite the limit:
\[ \lim_{x \to 0} \frac{\tan x}{x} = \lim_{x \to 0} \frac{\sec^2 x}{1} \]

### Step 3: Evaluate the limit
Now substitute \( x = 0 \) into \( \sec^2 x \):
\[ \sec^2(0) = \frac{1}{\cos^2(0)} = \frac{1}{1^2} = 1 \]

Thus, we have:
\[ \lim_{x \to 0} \frac{\tan x}{x} = 1 \]

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

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