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MCQOPTIONS
This section includes 8666 Mcqs, each offering curated multiple-choice questions to sharpen your Joint Entrance Exam - Main (JEE Main) knowledge and support exam preparation. Choose a topic below to get started.
| 3751. |
\[\underset{n\to \infty }{\mathop{\lim }}\,{{\left( \frac{n}{n+y} \right)}^{n}}\] equals [AMU 1999] |
| A. | 0 |
| B. | 1 |
| C. | 1/v |
| D. | \[{{e}^{-y}}\] |
| Answer» E. | |
| 3752. |
The value of \[\underset{a\to 0}{\mathop{\lim }}\,\frac{\sin a-\tan a}{{{\sin }^{3}}a}\]will be [UPSEAT 1999] |
| A. | \[-\frac{1}{2}\] |
| B. | \[\frac{1}{2}\] |
| C. | 1 |
| D. | ?1 |
| Answer» B. \[\frac{1}{2}\] | |
| 3753. |
The value of \[\underset{x\to 0}{\mathop{\lim }}\,\frac{x\cos x-\log (1+x)}{{{x}^{2}}}\]is [RPET 1999] |
| A. | ½ |
| B. | 0 |
| C. | 1 |
| D. | None of these |
| Answer» B. 0 | |
| 3754. |
\[\underset{x\to 0}{\mathop{\lim }}\,\frac{{{e}^{\frac{1}{x}}}}{{{e}^{\left( \frac{1}{x}+1 \right)}}}=\] [DCE 1999] |
| A. | 0 |
| B. | 1 |
| C. | Does not exist |
| D. | None of these |
| Answer» E. | |
| 3755. |
\[\underset{x\to 4}{\mathop{\lim }}\,\left[ \frac{{{x}^{3/2}}-8}{x-4} \right]=\] [DCE 1999] |
| A. | 3/2 |
| B. | 3 |
| C. | 2/3 |
| D. | 1/3 |
| Answer» C. 2/3 | |
| 3756. |
\[x=1\] is equal to [SCRA 1996] |
| A. | \[\frac{2}{3}\] |
| B. | \[1\] |
| C. | 0 |
| D. | \[\infty \] |
| Answer» B. \[1\] | |
| 3757. |
\[\underset{x\to 0}{\mathop{\lim }}\,\sin \left( \frac{1}{x} \right)\] is [SCRA 1996] |
| A. | 0 |
| B. | 1 |
| C. | ?1 |
| D. | Does not exist |
| Answer» E. | |
| 3758. |
\[\underset{x\to 0}{\mathop{\lim }}\,\frac{{{\tan }^{-1}}x}{x}\]is [SCRA 1996] |
| A. | 0 |
| B. | \[\infty \] |
| C. | ?1 |
| D. | 1 |
| Answer» E. | |
| 3759. |
\[\underset{x\to 0}{\mathop{\lim }}\,\frac{{{e}^{1/x}}-1}{{{e}^{1/x}}+1}=\] |
| A. | 0 |
| B. | 1 |
| C. | ?1 |
| D. | Does not exist |
| Answer» E. | |
| 3760. |
The value of the limit of \[\frac{{{x}^{3}}-{{x}^{2}}-18}{x-3}\]as x tends to 3 is [SCRA 1996] |
| A. | 3 |
| B. | 9 |
| C. | 18 |
| D. | 21 |
| Answer» E. | |
| 3761. |
Let the function f be defined by the equation \[f(x)=\left\{ \begin{align} & 3x\ \ \ \ \ \ \text{if}\ 0\le x\le 1 \\ & 5-3x\ \ \text{if}\ \text{1} |
| A. | \[\underset{x\to 1}{\mathop{\lim }}\,f(x)=f(1)\] |
| B. | \[\underset{x\to 1}{\mathop{\lim }}\,f(x)=3\] |
| C. | \[\underset{x\to 1}{\mathop{\lim }}\,f(x)=2\] |
| D. | \[\underset{x\to 1}{\mathop{\lim }}\,f(x)\]does not exist |
| Answer» E. | |
| 3762. |
The value of the limit of \[\frac{{{x}^{3}}-8}{{{x}^{2}}-4}\]as x tends to 2 is [SCRA 1996] |
| A. | 3 |
| B. | \[\frac{3}{2}\] |
| C. | 1 |
| D. | 0 |
| Answer» B. \[\frac{3}{2}\] | |
| 3763. |
\[\underset{x\to \infty }{\mathop{\lim }}\,{{\left[ 1+\frac{1}{mx} \right]}^{x}}\]equal to [Kurukshetra CEE 1998] |
| A. | \[{{e}^{1/m}}\] |
| B. | \[{{e}^{-1/m}}\] |
| C. | \[{{e}^{m}}\] |
| D. | \[{{m}^{e}}\] |
| Answer» B. \[{{e}^{-1/m}}\] | |
| 3764. |
The value of \[\underset{x\to -\infty }{\mathop{\lim }}\,\frac{\sqrt{4{{x}^{2}}+5x+8}}{4x+5}\]is [Roorkee 1998] |
| A. | \[-1/2\] |
| B. | 0 |
| C. | \[1/2\] |
| D. | 1 |
| Answer» B. 0 | |
| 3765. |
\[\underset{x\to 0}{\mathop{\lim }}\,\frac{x{{e}^{x}}-\log (1+x)}{{{x}^{2}}}\] equals [RPET 1996] |
| A. | \[\frac{2}{3}\] |
| B. | \[\frac{1}{3}\] |
| C. | \[\frac{1}{2}\] |
| D. | \[\frac{3}{2}\] |
| Answer» E. | |
| 3766. |
If \[f(x)=\left\{ \begin{align} & x,\ \ \text{if }x\text{ is rational } \\ & -x,\ \text{if }x\text{ is irrational} \\ \end{align} \right.,\] then \[\underset{x\to 0}{\mathop{\lim }}\,f(x)\]is [Kurukshetra CEE 1998; UPSEAT 2004] |
| A. | Equal to 0 |
| B. | Equal to 1 |
| C. | Equal to ?1 |
| D. | Indeterminate |
| Answer» B. Equal to 1 | |
| 3767. |
\[\underset{x\to 0}{\mathop{\lim }}\,\frac{\sqrt{1-{{x}^{2}}}-\sqrt{1+{{x}^{2}}}}{{{x}^{2}}}\] is equal to [MP PET 1999] |
| A. | 1 |
| B. | ?1 |
| C. | ?2 |
| D. | 0 |
| Answer» C. ?2 | |
| 3768. |
If \[\underset{x\to 5}{\mathop{\lim }}\,\frac{{{x}^{k}}-{{5}^{k}}}{x-5}=500\], then the positve integral value of k is [MP PET 1998] |
| A. | 3 |
| B. | 4 |
| C. | 5 |
| D. | 6 |
| Answer» C. 5 | |
| 3769. |
\[\underset{x\to \infty }{\mathop{\lim }}\,(\sqrt{{{x}^{2}}+8x+3}-\sqrt{{{x}^{2}}+4x+3})=\] [MP PET 1997] |
| A. | 0 |
| B. | \[\infty \] |
| C. | 2 |
| D. | \[\frac{1}{2}\] |
| Answer» D. \[\frac{1}{2}\] | |
| 3770. |
If \[\underset{x\to 0}{\mathop{\lim }}\,kx\,\text{cosec}\,x=\underset{x\to 0}{\mathop{\lim }}\,x\,\text{cosec}\ kx\], then \[k=\] |
| A. | 1 |
| B. | ?1 |
| C. | \[\pm 1\] |
| D. | \[\pm \,2\] |
| Answer» D. \[\pm \,2\] | |
| 3771. |
If \[f(x)=\frac{\sin ({{e}^{x-2}}-1)}{\log (x-1)},\]then \[\underset{x\to 2}{\mathop{\lim }}\,f(x)\]is given by |
| A. | ?2 |
| B. | ?1 |
| C. | 0 |
| D. | 1 |
| Answer» E. | |
| 3772. |
\[\underset{x\to \infty }{\mathop{\lim }}\,\frac{(2x-3)(3x-4)}{(4x-5)(5x-6)}=\] [MP PET 1996] |
| A. | 0 |
| B. | 1/10 |
| C. | 1/5 |
| D. | 3/10 |
| Answer» E. | |
| 3773. |
\[\underset{x\to 0}{\mathop{\lim }}\,{{\left( \frac{1+5{{x}^{2}}}{1+3{{x}^{2}}} \right)}^{1/{{x}^{2}}}}=\] [IIT 1996; DCE 2001] |
| A. | \[{{e}^{2}}\] |
| B. | \[e\] |
| C. | \[{{e}^{-2}}\] |
| D. | \[{{e}^{-1}}\] |
| Answer» B. \[e\] | |
| 3774. |
\[\underset{x\to 0}{\mathop{\lim }}\,\left[ \frac{\sin (x+a)+\sin (a-x)-2\sin a}{x\sin x} \right]=\] |
| A. | \[\sin a\] |
| B. | \[\cos a\] |
| C. | \[-\sin a\] |
| D. | \[\frac{1}{2}\cos a\] |
| Answer» D. \[\frac{1}{2}\cos a\] | |
| 3775. |
\[\underset{x\to \pi /2}{\mathop{\lim }}\,\left[ x\tan x-\left( \frac{\pi }{2} \right)\sec x \right]=\] |
| A. | 1 |
| B. | ?1 |
| C. | 0 |
| D. | None of these |
| Answer» C. 0 | |
| 3776. |
The value of \[\underset{x\to a}{\mathop{\lim }}\,\frac{\log (x-a)}{\log ({{e}^{x}}-{{e}^{a}})}\]is |
| A. | 1 |
| B. | ?1 |
| C. | 0 |
| D. | None of these |
| Answer» B. ?1 | |
| 3777. |
The value of \[\underset{x\to 0}{\mathop{\lim }}\,\frac{{{(1+x)}^{1/x}}-e+\frac{1}{2}ex}{{{x}^{2}}}\]is [DCE 2001] |
| A. | \[\frac{11e}{24}\] |
| B. | \[\frac{-11e}{24}\] |
| C. | \[\frac{e}{24}\] |
| D. | None of these |
| Answer» B. \[\frac{-11e}{24}\] | |
| 3778. |
The value of \[\underset{x\to \infty }{\mathop{\lim }}\,\frac{\log x}{{{x}^{n}}},\ n>0\]is |
| A. | \[0\] |
| B. | \[1\] |
| C. | \[\frac{1}{n}\] |
| D. | \[\frac{1}{n!}\] |
| Answer» B. \[1\] | |
| 3779. |
The value of \[\underset{x\to {{0}^{+}}}{\mathop{\lim }}\,{{x}^{m}}{{(\log x)}^{n}},\ m,\ n\in N\]is |
| A. | 0 |
| B. | \[\frac{m}{n}\] |
| C. | \[mn\] |
| D. | None of these |
| Answer» B. \[\frac{m}{n}\] | |
| 3780. |
The value of \[\underset{x\to 0}{\mathop{\lim }}\,\frac{\sqrt{1-\cos {{x}^{2}}}}{1-\cos x}\]is |
| A. | \[\frac{1}{2}\] |
| B. | \[2\] |
| C. | \[\sqrt{2}\] |
| D. | None of these |
| Answer» D. None of these | |
| 3781. |
\[\underset{x\to 1}{\mathop{\lim }}\,\frac{(2x-3)(\sqrt{x}-1)}{2{{x}^{2}}+x-3}=\] [IIT 1977] |
| A. | ?1/10 |
| B. | 1/10 |
| C. | ?1/8 |
| D. | None of these |
| Answer» B. 1/10 | |
| 3782. |
The value of \[\underset{x\to 2}{\mathop{\lim }}\,\frac{\sqrt{1+\sqrt{2+x}}-\sqrt{3}}{x-2}\]is |
| A. | \[\underset{x\to \frac{\pi }{2}}{\mathop{\lim }}\,\frac{\left[ 1-\tan \left( \frac{x}{2} \right) \right]\,[1-\sin x]}{\left[ 1+\tan \left( \frac{x}{2} \right) \right]\,{{[\pi -2x]}^{3}}}\] |
| B. | \[\frac{1}{4\sqrt{3}}\] |
| C. | 0 |
| D. | None of these |
| Answer» B. \[\frac{1}{4\sqrt{3}}\] | |
| 3783. |
The value of \[\underset{x\to 0}{\mathop{\lim }}\,{{\left( \frac{{{a}^{x}}+{{b}^{x}}+{{c}^{x}}}{3} \right)}^{2/x}}\]; \[(a,\ b,\ c>0)\] is |
| A. | \[{{(abc)}^{3}}\] |
| B. | \[abc\] |
| C. | \[{{(abc)}^{1/3}}\] |
| D. | None of these |
| Answer» E. | |
| 3784. |
If \[f(x)=\frac{2}{x-3},\ g(x)=\frac{x-3}{x+4}\] and \[h(x)=-\frac{2(2x+1)}{{{x}^{2}}+x-12},\] then \[\underset{x\to 3}{\mathop{\lim }}\,[f(x)+g(x)+h(x)]\] is |
| A. | \[-2\] |
| B. | \[-1\] |
| C. | \[-\frac{2}{7}\] |
| D. | 0 |
| Answer» D. 0 | |
| 3785. |
\[\underset{x\to \infty }{\mathop{\lim }}\,\left[ \sqrt{x+\sqrt{x+\sqrt{x}}}-\sqrt{x} \right]\]is equal to |
| A. | 0 |
| B. | \[\frac{1}{2}\] |
| C. | \[\frac{1}{p}-\frac{1}{p-1}\] |
| D. | \[{{e}^{4}}\] |
| Answer» C. \[\frac{1}{p}-\frac{1}{p-1}\] | |
| 3786. |
If \[f(x)=\sqrt{\frac{x-\sin x}{x+{{\cos }^{2}}x}}\], then \[\underset{x\to \infty }{\mathop{\lim }}\,f(x)\]is [DCE 2000] |
| A. | 0 |
| B. | \[\infty \] |
| C. | 1 |
| D. | None of these |
| Answer» D. None of these | |
| 3787. |
\[\underset{x\to 0}{\mathop{\lim }}\,\frac{{{e}^{\tan x}}-{{e}^{x}}}{\tan x-x}=\] [EAMCET 1994; RPET 2001] |
| A. | 1 |
| B. | \[e\] |
| C. | \[{{e}^{-1}}\] |
| D. | \[\frac{1}{2}\] |
| Answer» B. \[e\] | |
| 3788. |
The value of \[\underset{x\to \infty }{\mathop{\lim }}\,\sqrt{{{a}^{2}}{{x}^{2}}+ax+1}-\sqrt{{{a}^{2}}{{x}^{2}}+1}\]is |
| A. | \[\frac{1}{2}\] |
| B. | 1 |
| C. | \[2\] |
| D. | None of these |
| Answer» B. 1 | |
| 3789. |
If \[0 |
| A. | \[e\] |
| B. | \[x\] |
| C. | \[y\] |
| D. | None of these |
| Answer» D. None of these | |
| 3790. |
\[\underset{x\to 0}{\mathop{\lim }}\,{{\left\{ \tan \left( \frac{\pi }{4}+x \right) \right\}}^{1/x}}=\] [IIT 1993; RPET 2001] |
| A. | 1 |
| B. | ?1 |
| C. | \[{{e}^{2}}\] |
| D. | \[e\] |
| Answer» D. \[e\] | |
| 3791. |
\[\underset{x\to \infty }{\mathop{\lim }}\,{{\left( 1+\frac{2}{x} \right)}^{x}}=\] |
| A. | e |
| B. | \[\frac{1}{e}\] |
| C. | \[{{e}^{2}}\] |
| D. | None of these |
| Answer» D. None of these | |
| 3792. |
\[\underset{x\to \infty }{\mathop{\lim }}\,\sqrt{\frac{x+\sin x}{x-\cos x}}=\] [Roorkee 1994] |
| A. | 0 |
| B. | 1 |
| C. | ?1 |
| D. | None of these |
| Answer» C. ?1 | |
| 3793. |
The value of \[\underset{x\to \infty }{\mathop{\lim }}\,\frac{{{x}^{2}}\sin \frac{1}{x}-x}{1-|x|}\] is |
| A. | \[0\] |
| B. | 1 |
| C. | ?1 |
| D. | None of these |
| Answer» B. 1 | |
| 3794. |
\[\underset{n\to \infty }{\mathop{\lim }}\,{{({{4}^{n}}+{{5}^{n}})}^{1/n}}\]is equal to |
| A. | \[4\] |
| B. | 5 |
| C. | \[e\] |
| D. | None of these |
| Answer» C. \[e\] | |
| 3795. |
\[\underset{x\to 0}{\mathop{\lim }}\,{{\left( \frac{1+\tan x}{1+\sin x} \right)}^{\text{cosec }x}}\]is equal to [Kerala (Engg.) 2005] |
| A. | \[e\] |
| B. | \[\frac{1}{e}\] |
| C. | 1 |
| D. | None of these |
| Answer» D. None of these | |
| 3796. |
If \[a,\ b,\ c,\ d\] are positive, then \[\underset{x\to \infty }{\mathop{\lim }}\,{{\left( 1+\frac{1}{a+bx} \right)}^{c+dx}}=\] [EAMCET 1992] |
| A. | \[{{e}^{d/b}}\] |
| B. | \[{{e}^{c/a}}\] |
| C. | \[{{e}^{(c+d)/(a+b)}}\] |
| D. | \[e\] |
| Answer» B. \[{{e}^{c/a}}\] | |
| 3797. |
\[\underset{x\to 0}{\mathop{\lim }}\,\frac{\sin x+\log (1-x)}{{{x}^{2}}}\] is equal to [Roorkee 1995] |
| A. | 0 |
| B. | \[\frac{1}{2}\] |
| C. | \[-\frac{1}{2}\] |
| D. | None of these |
| Answer» D. None of these | |
| 3798. |
\[y\]exists, if [RPET 1995] |
| A. | \[{{x}_{n+1}}=\sqrt{2+{{x}_{n}}},\ n\ge 1,\ \] and \[\underset{x\to a}{\mathop{\lim }}\,g(x)\] exist |
| B. | \[\underset{x\to a}{\mathop{\lim }}\,f{{(x)}^{g(x)}}\] exists |
| C. | \[\underset{x\to a}{\mathop{\lim }}\,\frac{f(x)}{g(x)}\] exists |
| D. | \[\underset{x\to a}{\mathop{\lim }}\,f(x)g\left( \frac{1}{x} \right)\]exists |
| Answer» B. \[\underset{x\to a}{\mathop{\lim }}\,f{{(x)}^{g(x)}}\] exists | |
| 3799. |
\[\underset{x\to \frac{\pi }{2}}{\mathop{\lim }}\,|(1-\sin x)\tan x\] is |
| A. | \[\frac{\pi }{2}\] |
| B. | 1 |
| C. | 0 |
| D. | \[\infty \] |
| Answer» D. \[\infty \] | |
| 3800. |
\[\underset{x\to 0}{\mathop{\lim }}\,\frac{1-\cos 2x}{x}=\] [MNR 1983] |
| A. | 0 |
| B. | 1 |
| C. | 2 |
| D. | 4 |
| Answer» B. 1 | |