First slide

Introduction to limits

Question

Given $\lim_{x \to 0} \frac{f (x)}{x^{2}} = 2$ where [.] denotes the greatest integer function, then

Moderate

A

$\lim_{x \to 0} f [x] = 0$
B

$\lim_{x \to 0} f [x] = 1$
C

$\lim_{x \to 0} [\frac{f (x)}{x}]$ does not exist
D

$\lim_{x \to 0} [\frac{f (x)}{x}]$ exists

Solution

$\begin{array}{l} Since x^{2} > 0 and limit equala 2, f (x) must be a positive quantity . Also, since \lim_{x \to 0} \frac{f (x)}{x^{2}} = 2 ., \\ denominator \to zero and limit is finite . Therefore, f (x) must be approaching zero or \lim_{x \to 0} [f (x)] = 0^{+} . \end{array}$
$\begin{array}{l} Hence, \lim_{x \to 0} [f (x)] = 0^{+} . \\ \lim_{x \to 0^{+}} [\frac{f (x)}{x}] = \lim_{x \to 0^{+}} [x \frac{f (x)}{x^{2}}] = 0 \\ and \lim_{x \to 0^{-}} [\frac{f (x)}{x}] = \lim_{x \to 0^{-}} [x \frac{f (x)}{x^{2}}] = - 1 \\ Hence, \lim_{x \to 0} [\frac{f (x)}{x}] does not exist . \end{array}$

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