ₓ arrow 0} {9^{x}-4^{x}}{x≤ft(9^{x}+4^{x})}=

Correct Answer is : log⁡(32) ≤ft(3/2)log(23)

Correct Answer is : log⁡(32) ≤ft(3/2)log(23)

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MHTCET 10 May 2024 Shift 2 Paper

Question : 138 of 150

Marks: +1, -0

\lim _{x \rightarrow 0} \frac{9^{x}-4^{x}}{x\left(9^{x}+4^{x}\right)}=

Solution:

Concept:

Use the standard limit

\lim_{x \to 0} \frac{a^x - 1}{x} = \log_e a

Explanation:

Rewrite the numerator by adding and subtracting

1

9^x - 4^x = (9^x - 1) - (4^x - 1)

So the given limit becomes:

\lim_{x \to 0} \frac{(9^x - 1) - (4^x - 1)}{x} \times \frac{1}{9^x + 4^x}

Separate the terms in the first factor:

\left(\lim_{x \to 0} \frac{9^x - 1}{x} - \lim_{x \to 0} \frac{4^x - 1}{x}\right) \times \lim_{x \to 0} \frac{1}{9^x + 4^x}

Apply the standard formula:

(\log 9 - \log 4) \times \frac{1}{1+1}

Simplify:

\frac{1}{2} \log \frac{9}{4} = \frac{1}{2} \log \left(\frac{3}{2}\right)^2

= \log \frac{3}{2}

Answer:

\log \left(\frac{3}{2}\right)

Hence, option A is correct.

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