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: \[ \frac{1}{3} \]

Answer 2

Step 1:
Consider each term inside the limit separately and simplify using standard limit results.
We know that for small values of x, tan(x) ≈ x, hence the first term (tan(5x1/3)/5x1/3) → 1 as x → 0.

Step 2:
Similarly, the second term ((3√x)² / (tan⁻¹(3√x))²) → 1 because for small x, tan⁻¹(3√x) ≈ 3√x.

Step 3:
Now, (ℓ(1 + 3x²) / 3x²) → 1 as x → 0 because ℓ(1 + 3x²) ≈ 3x².

Step 4:
Also, (5x4/3 / (e5x4/3/3 - 1)) → 3 as x → 0, since the denominator approximates to 5x4/3/3 using the expansion ey - 1 ≈ y.

Step 5:
The last exponential factor (5x1/3·3x² / 5x4/3·9x) simplifies to 1/3.

Step 6:
Multiplying all the limits together, we get:
1 × 1 × 1 × 3 × (1/3) = 1/3.

Final Answer:
1/3