`lim_(x rarr 0) (int_(0)^(x) t tan(5t)dt)/(x^(3))` is equal to :

Options

A. `-(4)/(3)`
B. `-(5)/(3)`
C. `(4)/(3)`
D. `(5)/(3)`

Explanation

To solve the limit

\[ \lim_{x \to 0} \frac{\int_{0}^{x} t \tan(5t) \, dt}{x^3} \]

we will follow these steps:

### Step 1: Identify the form of the limit
As \( x \to 0 \), both the numerator and denominator approach 0. This gives us the indeterminate form \( \frac{0}{0} \).

**Hint:** Check if the limit results in an indeterminate form to decide if L'Hôpital's Rule is applicable.

### Step 2: Apply L'Hôpital's Rule
Since we have the \( \frac{0}{0} \) form, we can apply L'Hôpital's Rule, which involves taking the derivative of the numerator and the derivative of the denominator.

**Hint:** Remember that L'Hôpital's Rule states that if \(\lim_{x \to c} f(x) = \lim_{x \to c} g(x) = 0\), then \(\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}\).

### Step 3: Differentiate the numerator
Using Leibniz's rule for differentiation under the integral sign, we differentiate the numerator:

\[ \frac{d}{dx} \left( \int_{0}^{x} t \tan(5t) \, dt \right) = x \tan(5x) \]

**Hint:** When differentiating an integral with variable limits, use Leibniz's rule.

### Step 4: Differentiate the denominator
The derivative of the denominator \( x^3 \) is:

\[ \frac{d}{dx}(x^3) = 3x^2 \]

**Hint:** Remember the power rule for differentiation.

### Step 5: Rewrite the limit
Now we can rewrite the limit using the derivatives we calculated:

\[ \lim_{x \to 0} \frac{x \tan(5x)}{3x^2} \]

### Step 6: Simplify the limit
We can simplify this expression:

\[ \lim_{x \to 0} \frac{\tan(5x)}{3x} \]

### Step 7: Apply L'Hôpital's Rule again
As \( x \to 0 \), this limit is still in the \( \frac{0}{0} \) form. We apply L'Hôpital's Rule again:

The derivative of the numerator \( \tan(5x) \) is \( 5\sec^2(5x) \) and the derivative of the denominator \( 3x \) is \( 3 \):

\[ \lim_{x \to 0} \frac{5 \sec^2(5x)}{3} \]

### Step 8: Evaluate the limit
As \( x \to 0 \), \( \sec^2(5x) \to 1 \):

\[ \lim_{x \to 0} \frac{5 \cdot 1}{3} = \frac{5}{3} \]

### Final Answer
Thus, the limit is

\[ \frac{5}{3} \]