First slide

Introduction to limits

Question

The value of $\underset{x \to 1}{\lim_{_}} {\{\frac{x^{3} + 2 x^{2} + x + 1}{x^{2} + 2 x + 3}\}}^{\frac{1 - \cos (x - 1)}{(x - 1)^{2}}}$ , is

Moderate

A

$e$
B

$e^{1 / 2}$
C

1
D

none of these

Solution

We have,
$\begin{array}{l} lim_{x \to 1} {\{\frac{x^{3} + 2 x^{2} + x + 1}{x^{2} + 2 x + 3}\}}^{\frac{1 - \cos (x - 1)}{(x - 1)^{2}}} \\ = lim_{x \to 1} {\{\frac{x^{3} + 2 x^{2} + x + 1}{x^{2} + 2 x + 3}\}}^{\frac{2 \sin^{2} (x - 1) / 2}{(x - 1)^{2}}} \\ = {(\frac{5}{6})}^{\frac{2}{4}} = \sqrt{\frac{5}{6}} \end{array}$

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