First slide

Introduction to limits

Question

$I f \lim_{x \to a} \{\frac{f (x)}{g (x)}\}$ exists, then

Moderate

A

$both \lim_{x \to a} f (x) and \lim_{x \to a} g (x) must exist$
B

$both \lim_{x \to a} f (x) need not exist but \lim_{x \to a} g (x) exists$
C

$neither \lim_{x \to a} f (x) nor \lim_{x \to a} g (x) may exist$
D

$\lim_{x \to a} f (x) exists but \lim_{x \to a} g (x) need not exist$

Solution

$\begin{array}{l} If f (x) = \sin (\frac{1}{x}) and g (x) = \frac{1}{x}, then both \lim_{x \to 0} f (x) and \\ \lim_{x \to 0} g (x) do not exist, but \lim_{x \to 0} \frac{f (x)}{g (x)} = 0 exists . \end{array}$

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