First slide

Introduction to limits

Question

If $lim_{x \to 0} {(1 + x \log (1 + b^{2}))}^{1 / x} = 2 b \sin^{2} θ$ , $b > 0 and θ \in [- π, π]$ , then the value of $θ$ is

Moderate

A

$\frac{π}{4}$
B

$\frac{π}{3}$
C

$\frac{π}{2}$
D

$- \frac{π}{2}$

Solution

$\begin{array}{l} lim_{x \to 0} {(1 + x \log (1 + b^{2}))}^{1 / x} \\ = lim_{x \to 0} (1 + x \log {(1 + b^{2})}^{\frac{1}{x \log (1 + b^{2})} \times \log (1 + b^{2})} \\ = e^{\log (1 + b^{2})} = 1 + b^{2} \end{array}$
Thus ,
$1 + b^{2} = 2 b \sin^{2} θ$
$\begin{array}{l} \Rightarrow \sin^{2} θ = \frac{1 + b^{2}}{2 b} \geq 1 \\ \Rightarrow \sin^{2} θ = 1 \Rightarrow \sin θ = \pm 1 or θ = \pm π / 2 \end{array}$

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