Question

Evaluate the following limit: $ \lim_{x \to 0^+} \frac{\tan\left(5x^{\frac{1}{3}}\right) \log\left(1 + 3x^2\right)}{\left(\tan^{-1}\left(3\sqrt{x}\right)\right)^2 \left(e^{5x^{\frac{4}{3}}} - 1\right)} $

• A. \( \frac{1}{15} \)
• B. \( 1 \)
• C. \( \frac{1}{3} \)
• D. \( \frac{5}{3} \)

Hint:

When dealing with limits involving small angle approximations, use the standard expansions for \( \tan x \), \( \tan^{-1} x \), and \( e^x - 1 \) for small \( x \) to simplify the expression.

Answer 1

The Correct Option is C

Step 1: Approximate the functions for small values of \( x \).
We apply small angle approximations: \( \tan(x) \approx x \) as \( x \to 0 \), \( \tan^{-1}(x) \approx x \) as \( x \to 0 \), \( \log(1 + 3x^2) \approx 3x^2 \), \( e^x - 1 \approx x \) for small \( x \). 
Substituting in the expression:
\( \tan\left(5x^{\frac{1}{3}}\right) \approx 5x^{\frac{1}{3}} \), \( \log\left(1 + 3x^2\right) \approx 3x^2 \), \( \tan^{-1}\left(3\sqrt{x}\right) \approx 3\sqrt{x} \), \( e^{5x^{\frac{4}{3}}} - 1 \approx 5x^{\frac{4}{3}} \). 
Substitute these approximations into the given expression: \[ \frac{5x^{\frac{1}{3}} \cdot 3x^2}{(3\sqrt{x})^2 \cdot 5x^{\frac{4}{3}}} \] 
Step 2: Simplifying the expression.
Simplify the numerator and denominator: \[ \frac{15x^{\frac{7}{3}}}{9x \cdot 5x^{\frac{4}{3}}} \] This simplifies to: \[ \frac{15x^{\frac{7}{3}}}{45x^{\frac{7}{3}}} \] 
Step 3: Taking the limit as \( x \to 0 \)
. As \( x \to 0 \), the expression simplifies to: \[ \frac{15}{45} = \frac{1}{3} \] 
Thus, the final answer is: \[ \frac{1}{3} \]