`lim_(x to 0) (tan x - sinx)/x^(3)` =

Options

A. 1
B. 2
C. `1/2`
D. `-1`

Answer

To solve the limit \( \lim_{x \to 0} \frac{\tan x - \sin x}{x^3} \), we can follow these steps:

### Step 1: Rewrite \(\tan x\)
We start by rewriting \(\tan x\) in terms of \(\sin x\) and \(\cos x\):
\[ \tan x = \frac{\sin x}{\cos x} \]
So, we can express the limit as:
\[ \lim_{x \to 0} \frac{\frac{\sin x}{\cos x} - \sin x}{x^3} \]

### Step 2: Combine the terms in the numerator
Now, we can combine the terms in the numerator:
\[ \frac{\sin x}{\cos x} - \sin x = \sin x \left(\frac{1}{\cos x} - 1\right) = \sin x \left(\frac{1 - \cos x}{\cos x}\right) \]
Thus, the limit becomes:
\[ \lim_{x \to 0} \frac{\sin x (1 - \cos x)}{x^3 \cos x} \]

### Step 3: Use the small angle approximation for \(\sin x\) and \(\cos x\)
As \(x\) approaches 0, we can use the approximations:
\[ \sin x \approx x \quad \text{and} \quad 1 - \cos x \approx \frac{x^2}{2} \]
Substituting these approximations into our limit gives:
\[ \lim_{x \to 0} \frac{x \cdot \frac{x^2}{2}}{x^3 \cos x} \]

### Step 4: Simplify the expression
This simplifies to:
\[ \lim_{x \to 0} \frac{\frac{x^3}{2}}{x^3 \cos x} = \lim_{x \to 0} \frac{1}{2 \cos x} \]

### Step 5: Evaluate the limit
Now, as \(x\) approaches 0, \(\cos x\) approaches 1:
\[ \lim_{x \to 0} \frac{1}{2 \cos x} = \frac{1}{2 \cdot 1} = \frac{1}{2} \]

### Final Answer
Thus, the limit is:
\[ \lim_{x \to 0} \frac{\tan x - \sin x}{x^3} = \frac{1}{2} \]

---