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Question #40648

Difficulty: 8/10 | 🟢 Free

The Question

Let $L = \lim_{x \to \infty} \left( \frac{\sqrt[3]{x^3 + 162 x^2}}{\sqrt[4]{x^4 + 152 x^3}} - \frac{16}{x} \right)^{x^2}$ . Find the value of $\ln L$ .

Options

A.
$-1358$
B.
$694$
C.
$-1558$
D.
$-2916$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Binomial Theorem for Negative/Fractional Indices|Homogeneous Function Homogeneity|Infinite Limit Behavior|The 1^∞ Form Shortcut

Why This Question Is Tricky

Common Mistakes

This problem tests the rigorous application of the binomial theorem for fractional and negative indices within the context of a $1^\infty$ indeterminate limit.

The trap lies in the near-cancellation of the first-order terms, which forces students to expand up to the second-order ( $x^{-2}$ ) terms.

A common lethal mistake is either ignoring the cross-multiplication term ( $-2052x^{-2}$ ) when multiplying the two binomial expansions or miscalculating the second-order coefficients due to the negative fractional index ( $-1/4$ ).

By subtracting $16/x$ , the limit is perfectly isolated to the second-order coefficient, ruthlessly penalizing any algebraic shortcut.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Binomial Theorem - Class 11
Concept	Binomial Theorem for Negative/Fractional Indices
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free