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JEE Main

Explore popular questions from Limit, Continuity and Differentiability for JEE Main. This collection covers Limit, Continuity and Differentiability previous year JEE Main questions hand picked by experienced teachers.

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Limit, Continuity and Differentiability

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Q 1. Let {tex} f : R \rightarrow R {/tex} be a continuous function defined by {tex} f ( x ) = \frac { 1 } { e ^ { x } + 2 e ^ { - x } } {/tex}
Statement-1: {tex} f ( c ) = \frac { 1 } { 3 } , {/tex} for some {tex} c \in R . {/tex}
Statement-2: {tex} 0 < f ( x ) \leq \frac { 1 } { 2 \sqrt { 2 } } , {/tex} for all {tex} x \in R {/tex} .

A

Statement-1 is true, Statement- 2 is true; Statement- 2 is not the correct explanation for Statement- 1

B

Statement-1 is true, Statement- 2 is false

C

Statement-1 is false, Statement- 2 is true

Statement-1 is true, Statement- 2 is true; Statement- 2 is the correct explanation for Statement- 1

Explanation



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Q 2. {tex}\underset{ x \rightarrow 2 } \lim \left( \frac { \sqrt { 1 - \cos [ 2 ( x - 2 ) ] } } { x - 2 } \right) {/tex}

A

Equals {tex} \sqrt { 2 } {/tex}

B

Equals {tex} - \sqrt { 2 } {/tex}

C

Equals {tex} \frac { 1 } { \sqrt { 2 } } {/tex}

Does not exist

Explanation


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Q 3. If {tex} f : R \rightarrow R {/tex} is a function defined by {tex} f ( x ) = [ x ] \cos \left( \frac { 2 x - 1 } { 2 } \right) \pi {/tex}, where {tex} [ x ] {/tex} denotes the greatest integer function, then {tex} f {/tex} is

Continuous for every real {tex} x {/tex}

B

Discontinuous only at {tex} x = 0 {/tex}

C

Discontinuous only at non-zero integral values of {tex} x {/tex}

D

Continuous only at {tex} x = 0 {/tex}

Explanation

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Q 4. {tex}\underset{ x \rightarrow 0 } \lim \frac { \sin \left( \pi \cos ^ { 2 } x \right) } { x ^ { 2 } } {/tex} is equal to

A

{tex} - \pi {/tex}

{tex} \pi {/tex}

C

{tex} \frac { \pi } { 2 } {/tex}

D

{tex}\small1{/tex}

Explanation

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Q 5. If {tex} f ( x ) {/tex} is continuous and {tex} f \left( \frac { 9 } { 2 } \right) = \frac { 2 } { 9 } , {/tex} then {tex} \lim _ { x \rightarrow 0 } f \left( \frac { 1 - \cos 3 x } { x ^ { 2 } } \right) {/tex} is equal to

A

{tex}9 / 2 {/tex}

{tex}2 / 9 {/tex}

C

{tex}0{/tex}

D

{tex}8 / 9 {/tex}

Explanation


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Q 6. If {tex}\underset{ x \rightarrow 2 } \lim \frac { \tan ( x - 2 ) \left[ x ^ { 2 } + ( k - 2 ) x - 2 k \right] } { x ^ { 2 } - 4 x + 4 } = 5 , {/tex} then {tex} k {/tex} is equal to

A

0

B

1

C

2

3

Explanation


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Q 7. {tex}\underset{ x \rightarrow 0 } \lim \frac { e ^ { x ^ { 2 } } - \cos x } { \sin ^ { 2 } x } {/tex} is equal to

A

{tex}\small 3{/tex}

{tex} \frac { 3 } { 2 } {/tex}

C

{tex} \frac { 5 } { 4 } {/tex}

D

{tex}\small 2{/tex}

Explanation

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Q 8. Let {tex} a , b \in \mathrm { R } , ( a \neq 0 ) . {/tex} If the function {tex} f {/tex} defined as {tex} f ( x ) = \left\{ \begin{array} { c c } { \frac { 2 x ^ { 2 } } { a } , } & { 0 \leq x < 1 } \\ { a , } & { 1 \leq x < \sqrt { 2 } } \\ { \frac { 2 b ^ { 2 } - 4 b } { x ^ { 3 } } , } & { \sqrt { 2 } \leq x < \infty } \end{array} \right. {/tex}
is continuous in the interval {tex} [ 0 , \infty ) , {/tex} then an ordered pair {tex} ( a , b ) {/tex} is

A

{tex} ( - \sqrt { 2 } , 1 - \sqrt { 3 } ) {/tex}

B

{tex} ( \sqrt { 2 } , - 1 + \sqrt { 3 } ) {/tex}

{tex} ( \sqrt { 2 } , 1 - \sqrt { 3 } ) {/tex}

D

{tex} ( - \sqrt { 2 } , 1 + \sqrt { 3 } ) {/tex}

Explanation






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Q 9. {tex} \underset{ x \rightarrow 0 }\lim (\frac { \tan 2 x - x } { 3 x - \sin x } )= {/tex}

A

0

B

1

1/2

D

1/3

Explanation


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Q 10. {tex} \underset{ x \rightarrow 0 }\lim \frac { x \cdot 2 ^ { x } - x } { 1 - \cos x } = {/tex}

A

{tex}0{/tex}

{tex} \log 4 {/tex}

C

{tex} \log 2 {/tex}

D

None of these

Explanation


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Q 11. For {tex} x \in R , \underset{ x \rightarrow \infty }\lim \left( \frac { x - 3 } { x + 2 } \right) ^ { x } {/tex} is equal to number, then {tex} a {/tex} is equal to

A

{tex}0 {/tex}

B

{tex} e ^ { - 1 } {/tex}

{tex} e ^ { - 5 } {/tex}

D

{tex} e ^ { 5 } {/tex}

Explanation


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Q 12. {tex}\underset{ x \rightarrow 0 } \lim \frac { e ^ { x } - e ^ { - x } } { \sin x } {/tex} is

A

0

B

1

2

D

Non-existent

Explanation




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Q 13. {tex} \underset{ x \rightarrow \pi / 6 }\lim \left[ \frac { 3 \sin x - \sqrt { 3 } \cos x } { 6 x - \pi } \right] = {/tex}

A

{tex} \sqrt { 3 } {/tex}

{tex}1 / \sqrt { 3 } {/tex}

C

{tex} - \sqrt { 3 } {/tex}

D

{tex} - 1 / \sqrt { 3 } {/tex}

Explanation

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Q 14. {tex}\underset{ x \rightarrow 0 } \lim \frac { \cos ( \sin x ) - 1 } { x ^ { 2 } } = {/tex}

A

{tex}1{/tex}

B

{tex} - 1 {/tex}

C

1{tex} / 2 {/tex}

{tex} - 1 / 2 {/tex}

Explanation

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Q 15. The value of the constant {tex} \alpha {/tex} and {tex} \beta {/tex} such that {tex}\underset{ x \rightarrow \infty } \lim \left( \frac { x ^ { 2 } + 1 } { x + 1 } - \alpha x - \beta \right) = 0 {/tex} are respectively

A

{tex} ( 1,1 ) {/tex}

B

{tex} ( - 1,1 ) {/tex}

{tex} ( 1 , - 1 ) {/tex}

D

{tex} ( 0,1 ) {/tex}

Explanation



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Q 16. If {tex} S _ { n } = \underset {k=1}{ \overset {n} \sum} {/tex} and {tex} \underset{ n \rightarrow \infty } \lim a _ { n } = a , {/tex} then {tex} \underset{ n \rightarrow \infty }\lim \frac { S _ { n + 1 } - S _ { n } } { \underset {k=1}{ \overset {n} \sum k } } {/tex} is equal to

{tex}0{/tex}

B

{tex} a {/tex}

C

{tex} \sqrt { 2 } a {/tex}

D

{tex}2 a {/tex}

Explanation

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Q 17. The value of {tex}\underset { n \rightarrow \infty } \lim \cos \left( \frac { x } { 2 } \right) \cos \left( \frac { x } { 4 } \right) \cos \left( \frac { x } { 8 } \right) \cdots \cos \left( \frac { x } { 2 ^ { n } } \right) {/tex} is

A

{tex}\small 1{/tex}

{tex} \frac { \sin x } { x } {/tex}

C

{tex} \frac { x } { \sin x } {/tex}

D

None of these

Explanation



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Q 18. {tex}\underset { n \rightarrow \infty }\lim \left\{ \frac { 1 } { n ^ { 2 } } + \frac { 2 } { n ^ { 2 } } + \frac { 3 } { n ^ { 2 } } + \dots + \frac { n } { n ^ { 2 } } \right\} {/tex} is

{tex}1 / 2 {/tex}

B

{tex}0{/tex}

C

{tex}1{/tex}

D

{tex} \infty {/tex}

Explanation


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Q 19. If {tex} x _ { n } = \frac { 1 - 2 + 3 - 4 + 5 - 6 + \cdots - 2 n } { \sqrt { n ^ { 2 } + 1 } + \sqrt { 4 n ^ { 2 } - 1 } } , {/tex} then {tex}\underset{ n \rightarrow \infty } \lim x _ { n } {/tex} is equal to

A

{tex} \frac { 1 } { 3 } {/tex}

{tex} - \frac { 2 } { 3 } {/tex}

C

{tex} \frac { 2 } { 3 } {/tex}

D

{tex}\small 1{/tex}

Explanation

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Q 20. The value of {tex}\underset{ n \rightarrow \infty } \lim \frac { 1 + 2 + 3 + \cdots + n } { n ^ { 2 } + 100 } {/tex} is equal to

A

{tex}\infty{/tex}

{tex} \frac { 1 } { 2 } {/tex}

C

{tex}2{/tex}

D

{tex}0{/tex}

Explanation


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Q 21. If {tex} f ( x ) = | x - 2 | , {/tex} then

A

{tex}\underset { x \rightarrow 2 + } \lim _f ( x ) \neq 0 {/tex}

B

{tex} \underset{ x \rightarrow 2 - }\lim f ( x ) \neq 0 {/tex}

C

{tex}\underset{ x \rightarrow 2 + } \lim f ( x ) \neq \lim _ { x \rightarrow 2 - } f ( x ) {/tex}

{tex} f ( x ) {/tex} is continuous at {tex} x = 2 {/tex}

Explanation

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Q 22. The function {tex} f ( x ) = \frac { \log ( 1 + a x ) - \log ( 1 - b x ) } { x } {/tex} is not defined at {tex} x = 0 . {/tex} The value which should be assigned to {tex} f {/tex} at {tex} x = 0 {/tex} so that it is continuous at {tex} x = 0 {/tex} is

A

{tex} a - b {/tex}

{tex} a + b {/tex}

C

{tex} \log a + \log b {/tex}

D

{tex} \log a - \log b {/tex}

Explanation

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Q 23. Let {tex} f ( x ) = \left\{ \begin{array} { l l } { x ^ { 2 } + k , } & { \text { when } x \geq 0 } \\ { - x ^ { 2 } - k , } & { \text { when } x < 0 } \end{array} . \text { If the function } f ( x ) \text { be } \right. {/tex}

0

B

1

C

2

D

-2

Explanation

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Q 24. If {tex} f ( x ) = \left\{ \begin{array} { c } { \frac { x ^ { 2 } - 4 x + 3 } { x ^ { 2 } - 1 }}&{ , \text { for } x \neq 1 } \\ { 2}&{ , \quad \text { for } x = 1 } \end{array} , \text { then } \right. {/tex}

A

{tex}\underset { x \rightarrow 1 + }\lim f ( x ) = 2 {/tex}

B

{tex}\underset{ x \rightarrow 1 - } \lim f ( x ) = 3 {/tex}

{tex} f ( x ) {/tex} is discontinuous at {tex} x = 1 {/tex}

D

None of these

Explanation



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Q 25. The limit of {tex} f ( x ) = x ^ { 2 } {/tex} as {tex} x {/tex} tends to zero equals

zero

B

one

C

two

D

three

Explanation