GTU IMP MCQ Material With Mock Test Material For Mathematics Chapter Limit | Limit in math book material
Limit mcq questions, answers with solutions
1)`\lim_{θ\rightarrow0}\frac{\sin5x}x`
Answer:5
Explanation: `5\lim_{θ\rightarrow0}\frac{\sinx}x = 5(1) = 5`
2)`\lim_{x\rightarrow0}\frac{\3^x - 1}x`
answer:`log_3e`
explanation:
`\Rightarrow\lim_{x\rightarrow0}\frac{\3^x - 1}x = log_ea`
`\Rightarrow\log_e3`
3)`\lim_{x\rightarrow0}\left(1+\frac1x\right)^x`
answer:e
4)`\lim_{x\rightarrow2}\frac{x^3-8}x-2`
answer: 12
explanation `\Rightarrow\lim_{x\rightarrow2}\frac{x^n -a^n}{x-a} = na^{n-1}`
n = 3 and a = 2
`\Rightarrow 3(2)^{3-1}`
` \Rightarrow 3(2)^2`
12
5)`lim_{x\rightarrow0}\frac{sin2x}x `
answer:2
explanation:
`\Rightarrow 2 lim_{x\rightarrow0}\frac{sinx}x`
`\Rightarrow`2(1)
`\Rightarrow`2
6)`lim_{θ\rightarrow0}\frac{sin mθ}θ`
answer:-m
7)`lim_{x\rightarrow0} (sec^2x-tan^2x) `
answer:
explanation
we know that if put limit in sum and answer will be not 0 the put direct limit
`\Rightarrow sec^2(0)-tan^2(0)`
`\Rightarrow 1-0`
`\Rightarrow 1`
8)`lim_{x\rightarrow0}\frac{x^2+x+1}{x+1}`
answer:1
explanation `\frac{0^2+0+1}{0+1} = \frac{1}1 = 1`
9) `\lim_{x\rightarrow1}\frac{x^2 - 2x +1}{x-1}`
answer:0
explanation we did not put direct limit. use factorisation method (see explained all types method )
`\lim_{x\rightarrow1}\frac{(x-1)(x-1)}{x-1}`
= 0
10)`\lim_{x\rightarrow2}[\frac{1}{x-2}- \frac{4}{x^2-4}]`
answer`\frac{1}4`
explanation
`\Rightarrow \lim_{x\rightarrow2}[\frac{1}{x-2}- \frac{4}{x^2-2^2}]`
`\Rightarrow \lim_{x\rightarrow2}[\frac{x-2-4}{(x+2)(x-2)}]`
cancel (x-2) in numerator and denominator and put limit
11)`lim_{x\rightarrow1}\frac{x^2+x+1}{x+1}`
answer:`\frac{3}{2}`
explanation `\frac{1^2+1+1}{1+1} = \frac{3}2`
12)`lim_{x\rightarrow1}\frac{x^2-4x+3}{x^2+2x-3}`
answer:-`\frac{1}{2}`
explanation
`\Rightarrow lim_{x\rightarrow1}` `\frac{(x -1)(x -3)}{(x -1)(x +3)}`
`\Rightarrow lim_{x\rightarrow1}` ` \frac{x -3}{x +3}`
`\Rightarrow \frac{1-3}{1+3}` `\Rightarrow \frac{-2}{4}` =`\Rightarrow\frac{-1}{2}`
13)`lim_{x\rightarrow0}\frac{e^x - 1}x`
answer:1
explanation
`\Rightarrow lim_{x\rightarrow0}\frac{e^x - 1}x` `\Rightarrow log_ea = 1`
14)`lim_{x\rightarrow\infty}` x[x`\sqrt[2]` -1]
answer:`log_e2`
explanation
`\Rightarrow lim_{x\rightarrow\infty}` x[x`\sqrt[2]` -1]
`\Rightarrow\lim_{x\rightarrow\infty}` `\frac{2^\frac{1}{x} -1}{\frac{1}{x}}`
Take `\frac{1}{x}` = h. When x`\rightarrow``\infty` ,h`\rightarrow`0
`\Rightarrow lim_{h\rightarrow 0}``\frac{2^h -1}{h}` = `log_e2`
15)`lim_{x\rightarrow0}\frac{6^x - 2^x}x`
answer:`log_e3`
explanation
`\Rightarrow lim_{x\rightarrow0}\frac{(6^x - 1) -(2^x - 1)}x`
`\Rightarrow lim_{x\rightarrow0}\frac{(6^x - 1)}x` - `\Rightarrow lim_{x\rightarrow0}\frac{(2^x - 1)}x` `\Rightarrow log_e6` - `log_e2`
`\Rightarrow log_e \frac{6}{2}` `\Rightarrow log_e3`
16)`lim_{x\rightarrow\infty}\frac{∑ n}{n^2 -n}`
answer:`\frac{1}{2}`
explanation
`\Rightarrow lim_{x\rightarrow\infty} \frac{\frac{n(n+1)}{2}}{n(n-1)}` `\Rightarrow lim_{x\rightarrow\infty} \frac{n+1}{2(n-1)}`
`\Rightarrow \frac{1}{2} lim_{x\rightarrow\infty} \frac{1 +\frac{1}{n}}{1 -\frac{1}{n}}`
`\Rightarrow \frac{1}{2} \cdot \frac{1+0}{1-0}` `\Rightarrow \frac{1}{2}`
17)`lim_{x\rightarrow0}\frac{(1+x)^5 -1 }x`
answer:5
explanation
`\Rightarrow lim_{x\rightarrow0}\frac{(1+x)^5 -1 }x`
Take 1+x = y. When x `\rightarrow`0,y`\rightarrow`1. Also x =y -1.
`\Rightarrow lim_{y\rightarrow1} \frac{y^5 -1^5}{y-1}` =`5(1)^4` =5
18)`lim_{\theta\rightarrow0}\frac{sin m\theta}{sin n\theta}`
answer:`\frac{m}{n}`
explanation
`\Rightarrow lim_{\theta\rightarrow0} \frac{\frac{sin m\theta}{m\theta} × m\theta}{\frac{sin n\theta}{n\theta} × n\theta}`
`\Rightarrow \frac{lim_{\theta\rightarrow0} \frac{sin m\theta}{m\theta}}{lim_{\theta\rightarrow0} \frac{sin n\theta}{n\theta}}`× `\frac{m}{n}`
`\Rightarrow\frac{1}{1}\times \frac{m}{n}`
`\Rightarrow\frac{m}{n}`
19)`lim_{x\rightarrow0}\frac{sin^-1 x}x`
answer:1
explanation
`\Rightarrow lim_{x\rightarrow0}\frac{sin^-1 x}x`
Take `sin^-1`x =`\theta`, `\theta``\in` [-`\frac{\pi}{2}, \frac{\pi}{2}`].
= sin `\theta` = x. When x `\rightarrow` 0 , `\theta` `\rightarrow` 0
`\Rightarrow lim_{\theta\rightarrow0} \frac{\theta}{sin \theta}` `= \frac{1}{lim_{\theta\rightarrow0} \frac{sin\theta}{\theta}}` `=\frac{1}{1}` `\Rightarrow -1`
20)`lim_{x\rightarrow0}\frac{2sinx -sin2x}{x^3}`
answer:1
explanation
`\Rightarrow lim_{x\rightarrow0}\frac{2sinx-2sinx cosx}{x^3}`
`\Rightarrow lim_{x\rightarrow0}\frac{2sinx (1-cosx)}{x^3}`
`\Rightarrow lim_{x\rightarrow0} \frac{2sinx(1-cosx)}{x^3} \times \frac{1+cosx}{1+cosx}`
`\Rightarrow lim_{x\rightarrow0}\frac{2sinx (1-cos^2 x)}{x^3 (1+cosx)}`
`\Rightarrow lim_{x\rightarrow0} \frac{2sin^3 x}{x^3 (1+cosx)}`
`\Rightarrow lim_{x\rightarrow0}(\frac{sinx}{x}^3) x lim_{x\rightarrow0} \frac{2}{1+cosx}`
`\Rightarrow(1)^3 x \frac{2}{1+1}`
`\Rightarrow 1`
21)`lim_{x\rightarrow1}\frac{e^2x -1}x` = __________.
answer:`log_ee`
22)`lim_{\theta \rightarrow0}\frac{\theta}{tan3\theta}` = __________.
answer:`\frac{1}{3}`
23)`lim_{x \rightarrow1}\frac{1}{x^x-1}`
answer:1
=`\frac{1}{1^1-1}` =`\frac{1}{1}` =1
24)`lim_{y \rightarrow-1}\frac{y^15 +1}{y^11 +1}`
answer:`\frac{15}{11}`
`lim_{y \rightarrow-1}\frac{15y^14 +0}{11y^10 +0}`
`=\frac{15(-1)^14}{11(-1)^10}`
= `\frac{15}{11}`
25)`lim_{x \rightarrow0}\frac{sin(x^2)}{x}`
answer:0
= `lim_{x \rightarrow0}(\frac{sinx^2}{x^2}) (\frac{x^2}{x})`
= `lim_{x \rightarrow0}(\frac{sinx^2}{x^2})` (x)
=(1)(0) =0
26)`lim_{y \rightarrow-1}\frac{y^15 +1}{y^11 +1}`
answer:`\frac{15}{11}`
`lim_{y \rightarrow-1}\frac{15y^14 +0}{11y^10 +0}`
`=\frac{15(-1)^14}{11(-1)^10}`
= `\frac{15}{11}`
27)`lim_{x\rightarrow0}\frac{5^x-1}{x}`
Answer:`log_e5`
explanation:
`lim_{x\rightarrow0}\frac{a^x-1}{x} = log_ea `
` log_e5`
28)`lim_{x\rightarrow0}\frac{e^{2x} -1}{x}`
Answer:2
29)`lim_{x\rightarrow∞}(1+1/x)^x`
Answer:e
30)`lim_{x\rightarrow1}\frac{1}{x^{x-1}}`
Answer:1
31)`lim_{x\rightarrow0}\frac{sin3x}{2x}`
Answer:`3/2`
explanation:
`lim_{x\rightarrow0}\frac{sinx}{x}`
`3/2`
32)`lim_{x\rightarrow0}\frac{sin3x}{tan3x}`
Answer:3/5
explanation:
`3/5 lim_{x\rightarrow0}\frac{sinx\cdot cosx}{sinx}`
`3/5(1) = 3/5`
33)`lim_{θ\rightarrow0}\frac{1-cos θ}{ θ^2}`
Answer:1/2
explanation:
`\frac{1-cosθ}{θ^2}\times\frac{1+cosθ}{1-cosθ}`
`\frac{(1)^2 - (cosθ)^2}{θ^2(1+cosθ)}`
`\frac{1-cos^2θ}{θ^2(1+cosθ)}`
`\frac{sin^2θ}{θ^2} \times \frac{1}{1+cosθ}` `1/2`
34)`lim_{x\rightarrow0}\frac{a^{sinx}-1}{x}`
Answer:log_ea
35)`lim_{x\rightarrow0}\frac{sinx}{x}`
Answer:1
explanation:formula
Answer:5
Explanation: `5\lim_{θ\rightarrow0}\frac{\sinx}x = 5(1) = 5`
2)`\lim_{x\rightarrow0}\frac{\3^x - 1}x`
answer:`log_3e`
explanation:
`\Rightarrow\lim_{x\rightarrow0}\frac{\3^x - 1}x = log_ea`
`\Rightarrow\log_e3`
3)`\lim_{x\rightarrow0}\left(1+\frac1x\right)^x`
answer:e
4)`\lim_{x\rightarrow2}\frac{x^3-8}x-2`
answer: 12
explanation `\Rightarrow\lim_{x\rightarrow2}\frac{x^n -a^n}{x-a} = na^{n-1}`
n = 3 and a = 2
`\Rightarrow 3(2)^{3-1}`
` \Rightarrow 3(2)^2`
12
5)`lim_{x\rightarrow0}\frac{sin2x}x `
answer:2
explanation:
`\Rightarrow 2 lim_{x\rightarrow0}\frac{sinx}x`
`\Rightarrow`2(1)
`\Rightarrow`2
6)`lim_{θ\rightarrow0}\frac{sin mθ}θ`
answer:-m
7)`lim_{x\rightarrow0} (sec^2x-tan^2x) `
answer:
explanation
we know that if put limit in sum and answer will be not 0 the put direct limit
`\Rightarrow sec^2(0)-tan^2(0)`
`\Rightarrow 1-0`
`\Rightarrow 1`
8)`lim_{x\rightarrow0}\frac{x^2+x+1}{x+1}`
answer:1
explanation `\frac{0^2+0+1}{0+1} = \frac{1}1 = 1`
9) `\lim_{x\rightarrow1}\frac{x^2 - 2x +1}{x-1}`
answer:0
explanation we did not put direct limit. use factorisation method (see explained all types method )
`\lim_{x\rightarrow1}\frac{(x-1)(x-1)}{x-1}`
= 0
10)`\lim_{x\rightarrow2}[\frac{1}{x-2}- \frac{4}{x^2-4}]`
answer`\frac{1}4`
explanation
`\Rightarrow \lim_{x\rightarrow2}[\frac{1}{x-2}- \frac{4}{x^2-2^2}]`
`\Rightarrow \lim_{x\rightarrow2}[\frac{x-2-4}{(x+2)(x-2)}]`
cancel (x-2) in numerator and denominator and put limit
11)`lim_{x\rightarrow1}\frac{x^2+x+1}{x+1}`
answer:`\frac{3}{2}`
explanation `\frac{1^2+1+1}{1+1} = \frac{3}2`
12)`lim_{x\rightarrow1}\frac{x^2-4x+3}{x^2+2x-3}`
answer:-`\frac{1}{2}`
explanation
`\Rightarrow lim_{x\rightarrow1}` `\frac{(x -1)(x -3)}{(x -1)(x +3)}`
`\Rightarrow lim_{x\rightarrow1}` ` \frac{x -3}{x +3}`
`\Rightarrow \frac{1-3}{1+3}` `\Rightarrow \frac{-2}{4}` =`\Rightarrow\frac{-1}{2}`
13)`lim_{x\rightarrow0}\frac{e^x - 1}x`
answer:1
explanation
`\Rightarrow lim_{x\rightarrow0}\frac{e^x - 1}x` `\Rightarrow log_ea = 1`
14)`lim_{x\rightarrow\infty}` x[x`\sqrt[2]` -1]
answer:`log_e2`
explanation
`\Rightarrow lim_{x\rightarrow\infty}` x[x`\sqrt[2]` -1]
`\Rightarrow\lim_{x\rightarrow\infty}` `\frac{2^\frac{1}{x} -1}{\frac{1}{x}}`
Take `\frac{1}{x}` = h. When x`\rightarrow``\infty` ,h`\rightarrow`0
`\Rightarrow lim_{h\rightarrow 0}``\frac{2^h -1}{h}` = `log_e2`
15)`lim_{x\rightarrow0}\frac{6^x - 2^x}x`
answer:`log_e3`
explanation
`\Rightarrow lim_{x\rightarrow0}\frac{(6^x - 1) -(2^x - 1)}x`
`\Rightarrow lim_{x\rightarrow0}\frac{(6^x - 1)}x` - `\Rightarrow lim_{x\rightarrow0}\frac{(2^x - 1)}x` `\Rightarrow log_e6` - `log_e2`
`\Rightarrow log_e \frac{6}{2}` `\Rightarrow log_e3`
16)`lim_{x\rightarrow\infty}\frac{∑ n}{n^2 -n}`
answer:`\frac{1}{2}`
explanation
`\Rightarrow lim_{x\rightarrow\infty} \frac{\frac{n(n+1)}{2}}{n(n-1)}` `\Rightarrow lim_{x\rightarrow\infty} \frac{n+1}{2(n-1)}`
`\Rightarrow \frac{1}{2} lim_{x\rightarrow\infty} \frac{1 +\frac{1}{n}}{1 -\frac{1}{n}}`
`\Rightarrow \frac{1}{2} \cdot \frac{1+0}{1-0}` `\Rightarrow \frac{1}{2}`
17)`lim_{x\rightarrow0}\frac{(1+x)^5 -1 }x`
answer:5
explanation
`\Rightarrow lim_{x\rightarrow0}\frac{(1+x)^5 -1 }x`
Take 1+x = y. When x `\rightarrow`0,y`\rightarrow`1. Also x =y -1.
`\Rightarrow lim_{y\rightarrow1} \frac{y^5 -1^5}{y-1}` =`5(1)^4` =5
18)`lim_{\theta\rightarrow0}\frac{sin m\theta}{sin n\theta}`
answer:`\frac{m}{n}`
explanation
`\Rightarrow lim_{\theta\rightarrow0} \frac{\frac{sin m\theta}{m\theta} × m\theta}{\frac{sin n\theta}{n\theta} × n\theta}`
`\Rightarrow \frac{lim_{\theta\rightarrow0} \frac{sin m\theta}{m\theta}}{lim_{\theta\rightarrow0} \frac{sin n\theta}{n\theta}}`× `\frac{m}{n}`
`\Rightarrow\frac{1}{1}\times \frac{m}{n}`
`\Rightarrow\frac{m}{n}`
19)`lim_{x\rightarrow0}\frac{sin^-1 x}x`
answer:1
explanation
`\Rightarrow lim_{x\rightarrow0}\frac{sin^-1 x}x`
Take `sin^-1`x =`\theta`, `\theta``\in` [-`\frac{\pi}{2}, \frac{\pi}{2}`].
= sin `\theta` = x. When x `\rightarrow` 0 , `\theta` `\rightarrow` 0
`\Rightarrow lim_{\theta\rightarrow0} \frac{\theta}{sin \theta}` `= \frac{1}{lim_{\theta\rightarrow0} \frac{sin\theta}{\theta}}` `=\frac{1}{1}` `\Rightarrow -1`
20)`lim_{x\rightarrow0}\frac{2sinx -sin2x}{x^3}`
answer:1
explanation
`\Rightarrow lim_{x\rightarrow0}\frac{2sinx-2sinx cosx}{x^3}`
`\Rightarrow lim_{x\rightarrow0}\frac{2sinx (1-cosx)}{x^3}`
`\Rightarrow lim_{x\rightarrow0} \frac{2sinx(1-cosx)}{x^3} \times \frac{1+cosx}{1+cosx}`
`\Rightarrow lim_{x\rightarrow0}\frac{2sinx (1-cos^2 x)}{x^3 (1+cosx)}`
`\Rightarrow lim_{x\rightarrow0} \frac{2sin^3 x}{x^3 (1+cosx)}`
`\Rightarrow lim_{x\rightarrow0}(\frac{sinx}{x}^3) x lim_{x\rightarrow0} \frac{2}{1+cosx}`
`\Rightarrow(1)^3 x \frac{2}{1+1}`
`\Rightarrow 1`
21)`lim_{x\rightarrow1}\frac{e^2x -1}x` = __________.
answer:`log_ee`
22)`lim_{\theta \rightarrow0}\frac{\theta}{tan3\theta}` = __________.
answer:`\frac{1}{3}`
23)`lim_{x \rightarrow1}\frac{1}{x^x-1}`
answer:1
=`\frac{1}{1^1-1}` =`\frac{1}{1}` =1
24)`lim_{y \rightarrow-1}\frac{y^15 +1}{y^11 +1}`
answer:`\frac{15}{11}`
`lim_{y \rightarrow-1}\frac{15y^14 +0}{11y^10 +0}`
`=\frac{15(-1)^14}{11(-1)^10}`
= `\frac{15}{11}`
25)`lim_{x \rightarrow0}\frac{sin(x^2)}{x}`
answer:0
= `lim_{x \rightarrow0}(\frac{sinx^2}{x^2}) (\frac{x^2}{x})`
= `lim_{x \rightarrow0}(\frac{sinx^2}{x^2})` (x)
=(1)(0) =0
26)`lim_{y \rightarrow-1}\frac{y^15 +1}{y^11 +1}`
answer:`\frac{15}{11}`
`lim_{y \rightarrow-1}\frac{15y^14 +0}{11y^10 +0}`
`=\frac{15(-1)^14}{11(-1)^10}`
= `\frac{15}{11}`
27)`lim_{x\rightarrow0}\frac{5^x-1}{x}`
Answer:`log_e5`
explanation:
`lim_{x\rightarrow0}\frac{a^x-1}{x} = log_ea `
` log_e5`
28)`lim_{x\rightarrow0}\frac{e^{2x} -1}{x}`
Answer:2
29)`lim_{x\rightarrow∞}(1+1/x)^x`
Answer:e
30)`lim_{x\rightarrow1}\frac{1}{x^{x-1}}`
Answer:1
31)`lim_{x\rightarrow0}\frac{sin3x}{2x}`
Answer:`3/2`
explanation:
`lim_{x\rightarrow0}\frac{sinx}{x}`
`3/2`
32)`lim_{x\rightarrow0}\frac{sin3x}{tan3x}`
Answer:3/5
explanation:
`3/5 lim_{x\rightarrow0}\frac{sinx\cdot cosx}{sinx}`
`3/5(1) = 3/5`
33)`lim_{θ\rightarrow0}\frac{1-cos θ}{ θ^2}`
Answer:1/2
explanation:
`\frac{1-cosθ}{θ^2}\times\frac{1+cosθ}{1-cosθ}`
`\frac{(1)^2 - (cosθ)^2}{θ^2(1+cosθ)}`
`\frac{1-cos^2θ}{θ^2(1+cosθ)}`
`\frac{sin^2θ}{θ^2} \times \frac{1}{1+cosθ}` `1/2`
34)`lim_{x\rightarrow0}\frac{a^{sinx}-1}{x}`
Answer:log_ea
35)`lim_{x\rightarrow0}\frac{sinx}{x}`
Answer:1
explanation:formula
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