First slide

Introduction to limits

Question

The value of $lim_{x \to 3} \frac{3^{x} - x^{3}}{x^{x} - 3^{3}}$ , is

Moderate

A

$\frac{\log 3 - 1}{\log 3 + 1}$
B

$\frac{\log 3 + 1}{\log 3 - 1}$
C

1
D

none of these

Solution

we have,
$lim_{x \to 3} \frac{3^{x} - x^{3}}{x^{x} - 3^{2}}$
$= lim_{x \to 3} \frac{3^{x} \log_{e} 3 - 3 x^{2}}{x^{x} (1 + \log_{e} x) - 0}$ [Using L' Hospital's Rule]
$= \frac{3^{3} \log_{e} 3 - 3^{3}}{3^{3} (1 + \log_{e} 3)} = \frac{\log_{e} 3 - 1}{\log_{e} 3 + 1}$

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