First slide

Introduction to limits

Question

Given a real valued function f such that $f (x) = {\begin{matrix} \frac{\tan^{2} {x}}{(x^{2} - {[x]}^{2})} & f o r x > 0 \\ 1 & f o r x = 0 \\ \sqrt{{x} \cot {x}} & f o r x < 0 \end{matrix}$ , where [x] is the integral part and {x} is the fractional part of x then

Moderate

A

$\lim_{x \to 0^{+}} f (x) = 1$
B

$\lim_{x \to 0^{-}} f (x) = \sqrt{\cot} 1$
C

$\cot^{- 1} {(\lim_{x \to 0^{-}} f (x))}^{2} = 1$
D

$\tan^{- 1} (\lim_{x \to 0^{-}} f (x)) = \frac{π}{4}$

Solution

We have $\lim_{k \to 0^{+}} f (x) = \lim_{k \to 0^{+}} \frac{\tan^{2} {x}}{(x^{2} - {[x]}^{2})} = \lim_{x \to 0^{+}} \frac{\tan^{2} x}{x^{2}} = 1$ ……..(1) $[∵ x \to 0^{+}; [x] = 0 = {x} = x]$ Also $\lim_{k \to 0^{-}} f (x) = \lim_{x \to 0^{-}} \sqrt{{x} \cot {x}} = \sqrt{\cot 1} ..... (2)$ $[∵ x \to 0^{-}; [x] = - 1 \Rightarrow {x} = x + 1 \Rightarrow {x} \to 1]$ Also, $\cot^{- 1} {(\lim_{x \to 0^{-}} f (x))}^{2} = \cot^{- 1} (\cot 1) = 1$

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