Limits, Continuity and Differentiability Questions

MathematicsLimits, Continuity and Differentiability118 Questions Found

MathematicsRelationship Between Continuity and DifferentiabilityJEE Advanced 2007Moderate

In the following $[x]$ denotes the greatest integer less than or equal to $x$ . Match the functions in Column I with the properties in Column II and indicate your answer by darkening the appropriate bubbles in the $4 \times 4$ matrix given in the ORS.

List-I

(P)

x ∣ x ∣

(Q)

∣ x ∣

(R)

x + [x]

(S)

∣ x - 1∣ + ∣ x + 1∣

List-II

(1)

continuous in

(- 1, 1)

(2)

differentiable in

(- 1, 1)

(3)

strictly increasing in

(- 1, 1)

(4)

not differentiable at least at one point in

(- 1, 1)

Answer:P → NaN, P → 4, R → 4, P → 2

MathematicsContinuity at a Point and in an IntervalJEE Advanced 1981Easy

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Let $f (x + y) = f (x) + f (y)$ for all $x$ and $y$ . If the function $f (x)$ is continuous at $x = 0$ , then show that $f (x)$ is continuous at all $x$ .

MathematicsDifferentiability of a FunctionJEE Advanced 2001Moderate

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Let $α \in R$ . Prove that a function $f : R \to R$ is differentiable at $α$ if and only if there is a function $g : R \to R$ which is continuous at $α$ and satisfies $f (x) - f (α) = g (x) (x - α)$ for all $x \in R$ .

MathematicsEvaluation of Limits & L'Hopital's RuleJEE Advanced 1978Easy

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Evaluate $lim_{x \to a} \frac{a + 2 x - 3 x}{3 a + x - 2 x}, (a \neq = 0) .$

Answer:

\frac{2}{3 3}

MathematicsDifferentiability of a FunctionJEE Advanced 1979Easy

View in:English Hindi

Find the derivative of $f (x) = {\frac{x - 1}{2 x ^{2} - 7 x + 5} - \frac{1}{3} when x \neq = 1 when x = 1$ at $x = 1$ .

Answer:

- 2/9

MathematicsRelationship Between Continuity and DifferentiabilityJEE Advanced 1981Moderate

View in:English Hindi

Let $f (x) = {(x - 1)^{2} sin \frac{1}{( x - 1 )} - ∣ x ∣ - 1 if x \neq = 1 if x = 1$ be a real-valued function. Then the set of points where $f (x)$ is not differentiable is .........

Answer:

{0}

MathematicsEvaluation of Limits & L'Hopital's RuleJEE Advanced 1982Easy

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Use the formula $lim_{x \to 0} \frac{a ^{x} - 1}{x} = ln a$ to find $lim_{x \to 0} \frac{2 ^{x} - 1}{( 1 + x ) ^{1/2} - 1}$

Answer:

2 ln 2

MathematicsRelationship Between Continuity and DifferentiabilityJEE Advanced 1983Moderate

View in:English Hindi

Let $f (x) = {\frac{x ^{2}}{2}, 2 x^{2} - 3 x + \frac{3}{2}, 0 \leq x < 1 1 \leq x \leq 2$ . Discuss the continuity of $f, f^{'}$ and $f^{''}$ on $[0, 2]$ .

Answer:

f, f^{'}

continuous;

f^{''}

discontinuous

MathematicsDifferentiability of a FunctionJEE Advanced 1983Moderate

View in:English Hindi

For the function $f (x) = {\frac{x}{1 + e ^{1/ x}}, 0, x \neq = 0 x = 0$ , the derivative from the right, $f^{'} (0^{+}) = .........$ , and the derivative from the left, $f^{'} (0^{-}) = .........$

Answer:0, 1

MathematicsContinuity at a Point and in an IntervalJEE Advanced 1983Moderate

View in:English Hindi

Let $f (x) = {1 + x, 0 \leq x \leq 2 3 - x, 2 < x \leq 3$ . Determine the form of $g (x) = f (f (x))$ and hence find the points of discontinuity of $g$ , if any.

Answer:

x = 1, 2

...12

List-I

List-II

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f(x+y)=f(x)+f(y) for all x and y. If the function f(x) is continuous at x=0, then show that f(x) is continuous at all x.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Evaluate limx→a​3a+x​−2x​a+2x​−3x​​,(a=0).

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Find the derivative of f(x)={2x2−7x+5x−1​−31​​when x=1when x=1​ at x=1.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f(x)={(x−1)2sin(x−1)1​−∣x∣−1​if x=1if x=1​ be a real-valued function. Then the set of points where f(x) is not differentiable is .........

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Use the formula limx→0​xax−1​=lna to find limx→0​(1+x)1/2−12x−1​

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f(x)={2x2​,2x2−3x+23​,​0≤x<11≤x≤2​. Discuss the continuity of f,f′ and f′′ on [0,2].

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } For the function f(x)={1+e1/xx​,0,​x=0x=0​, the derivative from the right, f′(0+)=........., and the derivative from the left, f′(0−)=.........

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f(x)={1+x,0≤x≤23−x,2<x≤3​. Determine the form of g(x)=f(f(x)) and hence find the points of discontinuity of g, if any.

List-I

List-II

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f(x+y)=f(x)+f(y) for all x and y. If the function f(x) is continuous at x=0, then show that f(x) is continuous at all x.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Evaluate limx→a​3a+x​−2x​a+2x​−3x​​,(a=0).

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Find the derivative of f(x)={2x2−7x+5x−1​−31​​when x=1when x=1​ at x=1.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f(x)={(x−1)2sin(x−1)1​−∣x∣−1​if x=1if x=1​ be a real-valued function. Then the set of points where f(x) is not differentiable is .........

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Use the formula limx→0​xax−1​=lna to find limx→0​(1+x)1/2−12x−1​

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f(x)={2x2​,2x2−3x+23​,​0≤x<11≤x≤2​. Discuss the continuity of f,f′ and f′′ on [0,2].

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } For the function f(x)={1+e1/xx​,0,​x=0x=0​, the derivative from the right, f′(0+)=........., and the derivative from the left, f′(0−)=.........

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f(x)={1+x,0≤x≤23−x,2<x≤3​. Determine the form of g(x)=f(f(x)) and hence find the points of discontinuity of g, if any.

Let $f (x + y) = f (x) + f (y)$ for all $x$ and $y$ . If the function $f (x)$ is continuous at $x = 0$ , then show that $f (x)$ is continuous at all $x$ .

Evaluate $lim_{x \to a} \frac{a + 2 x - 3 x}{3 a + x - 2 x}, (a \neq = 0) .$

Find the derivative of $f (x) = {\frac{x - 1}{2 x ^{2} - 7 x + 5} - \frac{1}{3} when x \neq = 1 when x = 1$ at $x = 1$ .

Let $f (x) = {(x - 1)^{2} sin \frac{1}{( x - 1 )} - ∣ x ∣ - 1 if x \neq = 1 if x = 1$ be a real-valued function. Then the set of points where $f (x)$ is not differentiable is .........

Use the formula $lim_{x \to 0} \frac{a ^{x} - 1}{x} = ln a$ to find $lim_{x \to 0} \frac{2 ^{x} - 1}{( 1 + x ) ^{1/2} - 1}$

Let $f (x) = {\frac{x ^{2}}{2}, 2 x^{2} - 3 x + \frac{3}{2}, 0 \leq x < 1 1 \leq x \leq 2$ . Discuss the continuity of $f, f^{'}$ and $f^{''}$ on $[0, 2]$ .

For the function $f (x) = {\frac{x}{1 + e ^{1/ x}}, 0, x \neq = 0 x = 0$ , the derivative from the right, $f^{'} (0^{+}) = .........$ , and the derivative from the left, $f^{'} (0^{-}) = .........$

Let $f (x) = {1 + x, 0 \leq x \leq 2 3 - x, 2 < x \leq 3$ . Determine the form of $g (x) = f (f (x))$ and hence find the points of discontinuity of $g$ , if any.

Let $f (x + y) = f (x) + f (y)$ for all $x$ and $y$ . If the function $f (x)$ is continuous at $x = 0$ , then show that $f (x)$ is continuous at all $x$ .

Evaluate $lim_{x \to a} \frac{a + 2 x - 3 x}{3 a + x - 2 x}, (a \neq = 0) .$

Find the derivative of $f (x) = {\frac{x - 1}{2 x ^{2} - 7 x + 5} - \frac{1}{3} when x \neq = 1 when x = 1$ at $x = 1$ .

Let $f (x) = {(x - 1)^{2} sin \frac{1}{( x - 1 )} - ∣ x ∣ - 1 if x \neq = 1 if x = 1$ be a real-valued function. Then the set of points where $f (x)$ is not differentiable is .........

Use the formula $lim_{x \to 0} \frac{a ^{x} - 1}{x} = ln a$ to find $lim_{x \to 0} \frac{2 ^{x} - 1}{( 1 + x ) ^{1/2} - 1}$

Let $f (x) = {\frac{x ^{2}}{2}, 2 x^{2} - 3 x + \frac{3}{2}, 0 \leq x < 1 1 \leq x \leq 2$ . Discuss the continuity of $f, f^{'}$ and $f^{''}$ on $[0, 2]$ .

For the function $f (x) = {\frac{x}{1 + e ^{1/ x}}, 0, x \neq = 0 x = 0$ , the derivative from the right, $f^{'} (0^{+}) = .........$ , and the derivative from the left, $f^{'} (0^{-}) = .........$

Let $f (x) = {1 + x, 0 \leq x \leq 2 3 - x, 2 < x \leq 3$ . Determine the form of $g (x) = f (f (x))$ and hence find the points of discontinuity of $g$ , if any.