First slide

Introduction to limits

Question

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

Difficult

A

$\pm \frac{π}{4}$
B

$\pm \frac{π}{3}$
C

$\pm \frac{π}{6}$
D

$\pm \frac{π}{2}$

Solution

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

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