Class 12 | Limits | Suggestions and Importants Questions, Prevoius Years Solutons | Important for JEE Mains, JEE Advanced, AIEEE, WBJEE, and others

Limits Important Questions for JEE Mains, Advanced, AIEEE

Q1. $\lim_{x\rightarrow 0}\frac {log_{e}(1+3x)-log_{e}(3-x)}{x}=?$[AIEEE 2003]

Ans:
Shortcut method:

$\lim_{x\rightarrow 0}\frac {log_{e}(1+ax)-log_{e}(a-x)}{x}=\frac{2}{a}$
Put $a=3$
$\lim_{x\rightarrow 0}\frac {log_{e}(1+3x)-log_{e}(3-x)}{x}=\frac{2}{3}$

Q2.$\lim_{x\rightarrow \infty} (1+\frac{1}{x})^{\frac{1}{x}} = ?$  [JEE 2007]

Ans: Shortcut method:

Applying Formula 10 

$=\lim_{x\rightarrow \infty} e^{(1+\frac{1}{x}-1)\frac{1}{x}}$

$=e^{\frac{1}{2}}$


Q3. $\lim_{x\rightarrow \infty} (\frac{x+6}{x+1})^{x+4} =?$ [JEE 2004, IIT 1990]


$= \lim_{x\rightarrow \infty} (1+\frac{5}{x+1})^{x+1+3}$


Now,

Case:1

=$\lim_{x+1\rightarrow \infty} (1+\frac{5}{x+1})^{x+1}$

=$e^5$


Case:2

$=\lim_{x\rightarrow \infty} (1+\frac{5}{x+1})^{3}$

$= 1$

So, $\small \lim_{x\rightarrow \infty} (\frac{x+6}{x+1})^{x+4}=3$


Q4.$\lim_{x\rightarrow 0} (\frac{2^x-1}{\sqrt {x+1} -1})$=?[JEE 2010, IIT 1982]

Ans:
Dividing by $\small x$ for this limit $\small \lim_{x\rightarrow 0} (\frac{2^x-1}{\sqrt {x+1} -1})$

Trick:

Case1:

$\lim_{x\rightarrow 0} \frac{2^x-1}{x}=log_{e}2$


Case2:

$\small \lim_{x\rightarrow 0} \frac{\sqrt {x+1} -1}{x+1-1} = \frac{1}{2}$


Ans: $\small log_{e}2$


Q5.$\small \lim_{x\rightarrow 0} \frac{xe^x-log(1+x)}{x^2}=?$[WBJEE]


Ans:
$= \lim_{x\rightarrow 0} \frac{xe^x-x+x-log(1+x)}{x^2}$
$\small \lim_{x\rightarrow 0} [\frac{xe^x-x}{x^2}+\frac{x-(x-\frac{x^2}{2}+\frac{x^3}{3}-..)}{x^2}]$
$= 1+\small \frac{1}{2}+0+0+....$

Q6.$\small \lim_{x\rightarrow \infty} (\frac{5x^2+1}{3x^2+1})^{\frac{1}{x^2}}$=? [IIT 1996, WBJEE 2010]

Ans:

=$\small e^{\lim_{x\rightarrow \infty} (\frac{5x^2+1}{3x^2+1}-1)^{\frac{1}{x^2}}}$ 

=$\small e^{\lim_{x\rightarrow \infty} (\frac{2}{3x^2+1})^{\frac{1}{x^2}}}$

=$\small e^2$


Q7.$\small \lim_{n\rightarrow \infty} (\frac{5^n+3^n}{5^n-3^n})$=?

a. 1

b. 2

c. $\small \frac{1}{2}$

d. $\small \frac{1}{3}$

Ans: (a)

Hint:

$\small \lim_{n\rightarrow \infty} (\frac{5^n+3^n}{5^n-3^n})$

=$\small \lim_{n\rightarrow \infty} \frac{1+(\frac{3}{5})^n}{1-(\frac{3}{5})^n}$

Since $\small \frac{3}{5}<1$, and $\small n\rightarrow \infty$, and $(\frac{3}{5})\rightarrow 0$

 Click Here to See Limits Formula 

Q8. $\small \lim_{n\rightarrow \infty} [cos\frac{x}{n}]^n$=?

a. -1

b. 0

c. 1

d. None of these

Ans: 1

Hint:

$\small \lim_{n\rightarrow \infty} [cos\frac{x}{n}]^n$

$\small \lim_{n\rightarrow \infty} [1-2 sin^2\frac{x}{2n}]^n$

Apply Formula 10, when $\small f(n)=\small 1-2 sin^2\frac{x}{2n},g(n)=n$

Find, $\small \lim_{n\rightarrow \infty}f(n)^{g(n)}$

Q9. $\small \lim_{x\rightarrow 0^+}\frac{e^{\left [ x \right ]+\left | x \right |}-1}{\left [ x \right ]+\left | x \right |}$=?

a. 0

b. -1

c. 1

Ans: 1


Q10.$\small \lim_{x\rightarrow 0} (\frac{sinx}{x})^{\frac{sinx}{x-sinx}}$=?
a. $\small e$

b.$\small e^{-1}$

c.$\small e^2$

d.$\small e^{-2}$


Ans: $\small e^{-1}$


Hint: Apply formula 10


Q11.$\small \lim_{x\rightarrow 0} [tan(\frac{\pi}{4}+x)]^{\frac{1}{x}}$=?

a.$\small e$

b.$\small e^3$

c.$\small e^2$

d.$\small e^{-1}$


Ans: $\small e^2$


Hint:
Formula 10


Q12.$\small \frac{sin|x|}{|x|}$=?

a.1

b.-1

c.0

d. Does not exist


Ans: (d)


Q13.$\small \lim_{x\rightarrow 0} \frac{e^{x^2}-cosx}{x^2}=?$


a. $\small \frac{1}{2}$

b. 1

c. $\small \frac{2}{3}$

d.$\small \frac{3}{2}$


Ans: (d)


Hint: L'Hospitals Rule $\small [\frac{0}{0}]$ form



Q145.$\small\lim_{x\rightarrow 0} \frac{27^x-9^x-3^x+1}{\sqrt 2 -\sqrt(1+cosx)}$=?


a. $\small 4\sqrt 2(log 3)^2$

b. $\small 8\sqrt 2(log 3)^2$

c. $\small 12\sqrt 2(log 3)^2$

d. $\small 2\sqrt 2(log 3)^2$


Ans:(b)


Q15.$\small \lim_{x\rightarrow 1}\frac{sin(e^{x-1}-1)}{log_{e}x}$=?

a. 0

b. $\small e$

c. $\small \frac {1}{e}$

d.$\small 1$

Ans: d

Hint: Let, $\small z=x-1$

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