Limits, Continuity and Differentiability Questions

MathematicsLimits, Continuity and Differentiability118 Questions Found

MathematicsEvaluation of Limits & L'Hopital's RuleJEE Advanced 1984Easy

$lim_{x \to 1} (1 - x) tan \frac{π x}{2} = .........$

Answer:

2/ π

MathematicsEvaluation of Limits & L'Hopital's RuleJEE Advanced 1987Easy

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$lim_{x \to - \infty} [\frac{x ^{4} s i n ( 1/ x ) + x ^{2}}{( 1 + ∣ x ∣ ^{3} )}] = .........$

Answer:

- 1

MathematicsRelationship Between Continuity and DifferentiabilityJEE Advanced 1989Moderate

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Draw a graph of the function $y = [x] + ∣1 - x ∣, - 1 \leq x \leq 3$ . Determine the points, if any, where this function is not differentiable.

Answer:

x = 0, 1, 2, 3

MathematicsContinuity at a Point and in an IntervalJEE Advanced 1989Easy

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Find the values of $a$ and $b$ so that the function $f (x) = ⎩ ⎨ ⎧ x + a 2 sin x, 2 x cot x + b, a cos 2 x - b sin x, 0 \leq x < π /4 π /4 \leq x \leq π /2 π /2 < x \leq π$ is continuous for $0 \leq x \leq π$ .

Answer:

a = π /6, b = - π /12

MathematicsEvaluation of Limits & L'Hopital's RuleJEE Advanced 1989Moderate

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$A B C$ is an isosceles triangle inscribed in a circle of radius $r$ . If $A B = A C$ and $h$ is the altitude from $A$ to $B C$ then the triangle $A B C$ has perimeter $P = 2 (2 r h - h^{2} + 2 h r)$ and area $A = .........$ also $lim_{h \to 0} \frac{A}{P ^{3}} = .........$

Answer:

h 2 r h - h^{2}, \frac{1}{128 r}

MathematicsEvaluation of Limits & L'Hopital's RuleJEE Advanced 1990Easy

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$lim_{x \to \infty} (\frac{x + 6}{x + 1})^{x + 4} = .........$

Answer:

e^{5}

MathematicsEvaluation of Limits & L'Hopital's RuleJEE Advanced 1993Easy

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Find $lim_{x \to 0} {tan (π /4 + x)}^{1/ x} .$

Answer:

e^{2}

MathematicsContinuity at a Point and in an IntervalJEE Advanced 1994Moderate

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Let $f (x) = ⎩ ⎨ ⎧ {1 + ∣ sin x ∣}^{a /∣ s i n x ∣}, b, e^{t a n 2 x / t a n 3 x}, - π /6 < x < 0 x = 0 0 < x < π /6$ . Determine $a$ and $b$ such that $f (x)$ is continuous at $x = 0$ .

Answer:

a = 2/3, b = e^{2/3}

MathematicsEvaluation of Limits & L'Hopital's RuleJEE Advanced 1996Easy

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$lim_{x \to 0} (\frac{1 + 5 x ^{2}}{1 + 3 x ^{2}})^{1/ x^{2}} = ...$

Answer:

e^{2}

MathematicsContinuity at a Point and in an IntervalJEE Advanced 1996Moderate

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Let $f (x) = [x] sin (\frac{π}{[ x + 1 ]})$ , where $[∙]$ denotes the greatest integer function. The domain of $f$ is... and the points of discontinuity of $f$ in the domain are.....

Answer:

(- \infty, - 1) \cup [0, \infty), I - {0}

...12

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } limx→1​(1−x)tan2πx​=.........

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } limx→−∞​[(1+∣x∣3)x4sin(1/x)+x2​]=.........

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Draw a graph of the function y=[x]+∣1−x∣,−1≤x≤3. Determine the points, if any, where this function is not differentiable.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Find the values of a and b so that the function f(x)=⎩⎨⎧​x+a2​sinx,2xcotx+b,acos2x−bsinx,​0≤x<π/4π/4≤x≤π/2π/2<x≤π​ is continuous for 0≤x≤π.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } limx→∞​(x+1x+6​)x+4=.........

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Find limx→0​{tan(π/4+x)}1/x.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f(x)=⎩⎨⎧​{1+∣sinx∣}a/∣sinx∣,b,etan2x/tan3x,​−π/6<x<0x=00<x<π/6​. Determine a and b such that f(x) is continuous at x=0.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } limx→0​(1+3x21+5x2​)1/x2=...

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f(x)=[x]sin([x+1]π​), where [∙] denotes the greatest integer function. The domain of f is... and the points of discontinuity of f in the domain are.....

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } limx→1​(1−x)tan2πx​=.........

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } limx→−∞​[(1+∣x∣3)x4sin(1/x)+x2​]=.........

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Draw a graph of the function y=[x]+∣1−x∣,−1≤x≤3. Determine the points, if any, where this function is not differentiable.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Find the values of a and b so that the function f(x)=⎩⎨⎧​x+a2​sinx,2xcotx+b,acos2x−bsinx,​0≤x<π/4π/4≤x≤π/2π/2<x≤π​ is continuous for 0≤x≤π.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } limx→∞​(x+1x+6​)x+4=.........

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Find limx→0​{tan(π/4+x)}1/x.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f(x)=⎩⎨⎧​{1+∣sinx∣}a/∣sinx∣,b,etan2x/tan3x,​−π/6<x<0x=00<x<π/6​. Determine a and b such that f(x) is continuous at x=0.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } limx→0​(1+3x21+5x2​)1/x2=...

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f(x)=[x]sin([x+1]π​), where [∙] denotes the greatest integer function. The domain of f is... and the points of discontinuity of f in the domain are.....

$lim_{x \to 1} (1 - x) tan \frac{π x}{2} = .........$

$lim_{x \to - \infty} [\frac{x ^{4} s i n ( 1/ x ) + x ^{2}}{( 1 + ∣ x ∣ ^{3} )}] = .........$

Draw a graph of the function $y = [x] + ∣1 - x ∣, - 1 \leq x \leq 3$ . Determine the points, if any, where this function is not differentiable.

Find the values of $a$ and $b$ so that the function $f (x) = ⎩ ⎨ ⎧ x + a 2 sin x, 2 x cot x + b, a cos 2 x - b sin x, 0 \leq x < π /4 π /4 \leq x \leq π /2 π /2 < x \leq π$ is continuous for $0 \leq x \leq π$ .

$lim_{x \to \infty} (\frac{x + 6}{x + 1})^{x + 4} = .........$

Find $lim_{x \to 0} {tan (π /4 + x)}^{1/ x} .$

Let $f (x) = ⎩ ⎨ ⎧ {1 + ∣ sin x ∣}^{a /∣ s i n x ∣}, b, e^{t a n 2 x / t a n 3 x}, - π /6 < x < 0 x = 0 0 < x < π /6$ . Determine $a$ and $b$ such that $f (x)$ is continuous at $x = 0$ .

$lim_{x \to 0} (\frac{1 + 5 x ^{2}}{1 + 3 x ^{2}})^{1/ x^{2}} = ...$

Let $f (x) = [x] sin (\frac{π}{[ x + 1 ]})$ , where $[∙]$ denotes the greatest integer function. The domain of $f$ is... and the points of discontinuity of $f$ in the domain are.....

$lim_{x \to 1} (1 - x) tan \frac{π x}{2} = .........$

$lim_{x \to - \infty} [\frac{x ^{4} s i n ( 1/ x ) + x ^{2}}{( 1 + ∣ x ∣ ^{3} )}] = .........$

Draw a graph of the function $y = [x] + ∣1 - x ∣, - 1 \leq x \leq 3$ . Determine the points, if any, where this function is not differentiable.

Find the values of $a$ and $b$ so that the function $f (x) = ⎩ ⎨ ⎧ x + a 2 sin x, 2 x cot x + b, a cos 2 x - b sin x, 0 \leq x < π /4 π /4 \leq x \leq π /2 π /2 < x \leq π$ is continuous for $0 \leq x \leq π$ .

$lim_{x \to \infty} (\frac{x + 6}{x + 1})^{x + 4} = .........$

Find $lim_{x \to 0} {tan (π /4 + x)}^{1/ x} .$

Let $f (x) = ⎩ ⎨ ⎧ {1 + ∣ sin x ∣}^{a /∣ s i n x ∣}, b, e^{t a n 2 x / t a n 3 x}, - π /6 < x < 0 x = 0 0 < x < π /6$ . Determine $a$ and $b$ such that $f (x)$ is continuous at $x = 0$ .

$lim_{x \to 0} (\frac{1 + 5 x ^{2}}{1 + 3 x ^{2}})^{1/ x^{2}} = ...$

Let $f (x) = [x] sin (\frac{π}{[ x + 1 ]})$ , where $[∙]$ denotes the greatest integer function. The domain of $f$ is... and the points of discontinuity of $f$ in the domain are.....