Exercise 27.1
Page no 27.21
Question 1
Evaluate the following limits :
(i) $\lim _{x \rightarrow 1}(x-2)$
(ii) $\lim _{x \rightarrow 2}(x+2)$
(iii) $\lim _{x \rightarrow 0}(9)$
(iv) $\lim _{x \rightarrow 3} \frac{x^{2}-9}{x+2}$
(v) $\lim _{x \rightarrow 2}(3-x)$
(vi) $\lim _{x \rightarrow 2} x(x-1)$
(vii) $\lim _{x \rightarrow-1} \frac{x^{3}-3 x+1}{x-1}$
(viii) $\lim _{x \rightarrow 2} \frac{x^{2}+4}{x+2}$
Page no 27.22
(ix) $\lim _{x \rightarrow 1} \frac{x^{2}+5}{x-2}$
(x) $\lim _{x \rightarrow 0} \frac{a x+b}{c x+d}, d \neq 0$
(xi) $\lim _{x \rightarrow 0} \frac{3 x+1}{x+3}$
(xii) $\lim _{x \rightarrow 3} \frac{x^{2}-4 x}{x-2}$
(xiii) $\lim _{r \rightarrow 1} \pi r^{2}$
(xiv) $\lim _{x \rightarrow 3} x(x+3)$
(xv) $\lim _{x \rightarrow 3} x+3$
(xvi) $\lim _{x \rightarrow 1}\left(x^{3}-x^{2}+1\right)$
(xvii) $\lim _{x \rightarrow \pi}\left(x-\frac{22}{7}\right)$
(xviii) $\lim _{x \rightarrow-1}\left(1+x+x^{2}+\ldots+x^{10}\right)$
(xix) $\lim _{x \rightarrow 0} \frac{a x+b}{c x+1}$
(xx) $\lim _{x \rightarrow 4} \frac{4 x+3}{x-2}$
(xxi) $\lim _{x \rightarrow 1}\left(\frac{x^{2}+1}{x+100}\right)$
(xxii) $\lim _{x \rightarrow 1} \frac{a x^{2}+b x+c}{c x^{2}+b x+a}, a+b+c \neq 0$
(xxiii) $\lim _{x \rightarrow-1} \frac{x^{10}+x^{5}+1}{x-1}$
Question 2
Evaluate $\lim _{x \rightarrow 1} \frac{x^{3}-1}{x-1}$
Question 3
Find $\lim _{x \rightarrow a} \frac{x^{4}-a^{4}}{x-a}$
Question 4
Find $\lim _{x \rightarrow 2} \frac{x^{4}-16}{x-2}$
Question 5
Find $\lim _{t \rightarrow-3} \frac{x^{3}+27}{x^{5}+243}$
Question 6
Find (i) $\lim _{t \rightarrow 1} \frac{\sqrt{t}-1}{\sqrt[3]{t}-1}$
Question 7
Find $\lim _{x \rightarrow a} \frac{x-a}{x^{3 / 2}-a^{3 / 2}}$
Question 8
Find $\lim _{x \rightarrow 1} \frac{x-1}{x^{1 / 4}-1}$
Question 9
Find $\lim _{x \rightarrow 2} \frac{\sqrt[3]{x-1}-1}{x-2}$
Question 10
Find $\lim _{x \rightarrow 0} \frac{(1+x)^{n}-1}{x}$
Question 11
Find $\lim _{x \rightarrow 1} \frac{x^{5}-1}{x-1}$
Question 12
Find $\lim _{x \rightarrow 0} \frac{(1+x)^{1 / n}-1}{x}$
Question 13
Find
$\lim _{x \rightarrow 0} \frac{(1+x)^{1 / 3}-(1-x)^{1 / 3}}{x}$
Question 14
Find $\lim _{x \rightarrow 1}\left(\frac{1-x^{-1 / 3}}{1-x^{-2 / 3}}\right)$
Question 15
Find $\lim _{x \rightarrow 0} \frac{(1+x)^{5}-1}{x}$
Question 16
Find $\lim _{x \rightarrow a} \frac{(x+2)^{5 / 2}-(a+2)^{5 / 2}}{x-a}$
Question 17
Find the positive integer $n$ so that $\lim _{x \rightarrow 3} \frac{x^{n}-3^{n}}{x-3}=108$
Question 18
Find all possible values of $n \in N$ if $\lim _{x \rightarrow 2} \frac{x^{n}-2^{n}}{x-2}=80$
Question 19
Find all possible values of $a$, if $\lim _{x \rightarrow a} \frac{x^{3}-a^{3}}{x-a}=\lim _{x \rightarrow 1} \frac{x^{4}-1}{x-1}$
Page no 27.23
Question 20
$\operatorname{Find}_{x \rightarrow 0} \frac{\sqrt{1+4 x}-\sqrt{1-3 x}}{x}$
Question 21
Find $\lim _{x \rightarrow 0} \frac{\sqrt{1+x^{2}}-\sqrt{1-x^{2}}}{x}$
Question 22
Find $\lim _{x \rightarrow 1} \frac{\sqrt{2-x}-1}{1-x}$
Question 23
Find $\lim _{x \rightarrow 0} \frac{\sqrt{1+x}-1}{x}$
Question 24
Find $\lim _{x \rightarrow 0} \frac{2-\sqrt{4-x^{2}}}{x^{2}}$
Question 25
Find $\lim _{x \rightarrow 0} \frac{a-\sqrt{a^{2}-x^{2}}}{x}$
Question 26
Find $\lim _{x \rightarrow 0} \frac{\sqrt{1+x}-\sqrt{1-x}}{x}$
Question 27
Find $\lim _{h \rightarrow 0} \frac{\sqrt{x+h}-\sqrt{x}}{h}, x \neq 0$
Question 28
Find $\lim _{x \rightarrow 0} \frac{\sqrt{a+x}-\sqrt{a}}{x \sqrt{a(a+x)}}$
Question 29
Find $\lim _{x \rightarrow 1} \frac{(2 x-3)(\sqrt{x}-1)}{2 x^{2}+x-3}$
Question 30
Find $\lim _{x \rightarrow 2} \frac{x^{2}-4}{\sqrt{x+2}-\sqrt{3 x-2}}$
Question 31
Find $\lim _{x \rightarrow 1} \frac{x-1}{\sqrt{x^{2}-1}+\sqrt{x-1}}$
Question 32
Find $\lim _{x \rightarrow 0} \frac{\sqrt{1+x^{2}}-\sqrt{1+x}}{\sqrt{x^{2}+1}+\sqrt{1+x}}$
Question 33
Find $\lim _{x \rightarrow 0} \frac{\sqrt{1+2 x^{2}}-\sqrt{1+x}}{\sqrt{4+4 x^{2}}-\sqrt{4-3 x}}$
Question 34
Find $\lim _{x \rightarrow 1} \frac{\sqrt{x^{2}+8}-\sqrt{10-x^{2}}}{\sqrt{x^{2}+3}-\sqrt{5-x^{2}}}$
Question 35
Find $\lim _{x \rightarrow 1} \frac{x^{2}-3 x+2}{x^{2}-1}$
Question 36
Find $\lim _{x \rightarrow 1} \frac{x^{2}-3 x+2}{x^{2}-5 x+4}$
Question 37
Find $\lim _{x \rightarrow 5} \frac{x^{2}-9 x+20}{x^{2}-6 x+5}$
Question 38
$\lim _{x \rightarrow-2}\left(\frac{\frac{1}{x}+\frac{1}{2}}{x+2}\right)$
Question 39
$\lim _{x \rightarrow 2} \frac{3 x^{2}-x-10}{x^{2}-4}$
Question 40
$\lim _{x \rightarrow 2} \frac{x^{3}-2 x^{2}}{x^{2}-5 x+6}$
Question 41
$\lim _{x \rightarrow 2} \frac{x^{3}-4 x^{2}+4 x}{x^{2}-4}$
Question 42
$\lim _{x \rightarrow 3} \frac{x^{4}-81}{2 x^{2}-5 x-3}$
Question 43
$\lim _{x \rightarrow 2} \frac{x^{2}-4}{x^{3}-4 x^{2}+4 x}$
Question 44
find $\lim _{x \rightarrow 2} \frac{2 x^{3}+3 x^{2}-8 x-12}{x^{3}+5 x^{2}-4 x-20}$
Question 45
If $f(x)=\frac{1}{x}$. prove that $\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}=-\frac{1}{x^{2}}$
Question 46
Find $\lim _{x \rightarrow 2}\left(\frac{1}{x-2}-\frac{4}{x^{2}-4}\right)$
Question 47
Find $\lim _{x \rightarrow 2}\left[\frac{1}{x-2}-\frac{2(2 x-3)}{x^{3}-3 x^{2}+2 x}\right]$
Question 48
$\lim _{x \rightarrow 1}\left[\frac{x-2}{x^{2}-x}-\frac{1}{x^{3}-3 x^{2}+2 x}\right]$
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