First slide

Introduction to limits

Question

$lim_{x \to 0} \frac{x (e^{\frac{\sqrt{1 + x^{2} + x^{4}} - 1}{x}} - 1)}{\sqrt{1 + x^{2} + x^{4} - 1}}$

Moderate

A

does not exist
B

is equal to 1
C

is equal
D

is equal to 0

Solution

We find that
$lim_{x \to 0} \frac{\sqrt{1 + x^{2} + x^{4}} - 1}{x} = lim_{x \to 0} \frac{x^{2} (1 + x^{2})}{x (\sqrt{1 + x^{2} + x^{4}} + 1)} = 0$
Let $y = \frac{\sqrt{1 + x^{2} + x^{4}} - 1}{x} .$ Then,
$∴ lim_{x \to 0} \frac{x (e^{\frac{\sqrt{1 + x^{2} + x^{4}} - 1}{x}} - 1)}{\sqrt{1 + x^{2} + x^{4}} - 1}$
$= lim_{x \to 0} \frac{(e^{\frac{\sqrt{1 + x^{2} + x^{4}} - 1}{x} - 1})}{\frac{\sqrt{1 + x^{2} + x^{4}} - 1}{x}} = lim_{y \to 0} \frac{e^{y} - 1}{y} = 1$

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