Match the following lists:
Column -I Column -II
A. If $L = lim_{x \to - 1} \frac{\sqrt[3]{(7 - x)} - 2}{(x + 1)}$ , then 12L = P. -2
B. If $L = lim_{x \to π / 4} \frac{\tan^{3} x - \tan x}{\cos (x + \frac{π}{4})}$ then -L/4= Q. 2
C. If $L = lim_{x \to 1} \frac{(2 x - 3) (\sqrt{x} - 1)}{2 x^{2} + x - 3}$ , then 20L= R. 1
D. If $L = lim_{x \to \infty} \frac{\log x^{n} - [x]}{[x]}$ , where $n \in N$
([x] denotes greatest integer less than or equal to x),then -2L=
S. -1

Moderate

Solution

$a. Let x + 1 = h . Then,$
$\begin{array}{l} lim_{x \to - 1} \frac{\sqrt[3]{(7 - x)} - 2}{(x + 1)} = lim_{h \to 0} \frac{(8 - h)^{1 / 3} - 2}{h} \\ = lim_{h \to 0} \frac{2 {(1 - \frac{h}{8})}^{1 / 3} - 2}{h} \\ = 2 lim_{h \to 0} \frac{(1 - \frac{1}{3} \frac{h}{8}) - 1}{h} \\ = - \frac{1}{12} \end{array}$
b. we have
$\begin{array}{l} lim_{x \to π / 4} \frac{\tan^{3} x - \tan x}{\cos (x + π / 4)} \\ = lim_{x \to π / 4} \frac{\tan x (\tan x - 1) (\tan x + 1)}{\cos (x + π / 4)} \\ = lim_{x \to π / 4} \frac{\tan x (\sin x - \cos x) (\tan x + 1)}{\cos xcos (x + π / 4)} \\ = - lim_{x \to π / 4} \frac{\tan x (\cos x - \sin x) (\tan x + 1)}{\cos xcos (x + π / 4)} \\ = - \sqrt{2} lim_{x \to π / 2} \frac{\tan x (\frac{1}{\sqrt{2}} \cos x - \frac{1}{\sqrt{2}} \sin x) (\tan x + 1)}{\cos xcos (x + π / 4)} \\ = - \sqrt{2} lim_{x \to π / 4} \frac{\tan x (\tan x + 1)}{\cos x} \\ = - \sqrt{2} \times 2 \times \sqrt{2} = - 4 \end{array}$
c.
$\begin{array}{l} lim_{x \to 1} \frac{(2 x - 3) (\sqrt{x} - 1)}{2 x^{2} + x - 3} = lim_{x \to 1} \frac{(2 x - 3) (\sqrt{x} - 1)}{(2 x + 3) (x - 1)} \\ = lim_{x \to 1} \frac{(2 x - 3) (\sqrt{x} - 1)}{(2 x + 3) (\sqrt{x} - 1) (\sqrt{x} + 1)} \\ = lim_{x \to 1} \frac{(2 x - 3)}{(2 x + 3) (\sqrt{x} + 1)} \\ = \frac{2 - 3}{(2 + 3) (\sqrt{1} + 1)} = - 1 / 10 \end{array}$
d.
$\begin{matrix} lim_{x \to \infty} \frac{\log x^{n} - [x]}{[x]} = lim_{x \to \infty} \frac{\log x^{n}}{[x]} - lim_{x \to \infty} \frac{[x]}{[x]} \\ = 0 - 1 = - 1 \end{matrix}$

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Column -I	Column -II
A. If $L = lim_{x \to - 1} \frac{\sqrt[3]{(7 - x)} - 2}{(x + 1)}$ , then 12L =	P. -2
B. If $L = lim_{x \to π / 4} \frac{\tan^{3} x - \tan x}{\cos (x + \frac{π}{4})}$ then -L/4=	Q. 2
C. If $L = lim_{x \to 1} \frac{(2 x - 3) (\sqrt{x} - 1)}{2 x^{2} + x - 3}$ , then 20L=	R. 1
D. If $L = lim_{x \to \infty} \frac{\log x^{n} - [x]}{[x]}$ , where $n \in N$ ([x] denotes greatest integer less than or equal to x),then -2L=	S. -1