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Questions › DSE › DSE 2010 › Q32
2 marksMathematics › Limits, continuity and differentiability › Differentiability and its link to continuityModerateabout 3 min

DSE 2010 Question 32: Limits, continuity and differentiability

Suppose a function f:ℜ→ℜf:\Re\to\Ref:ℜ→ℜ is differentiable at xxx. Consider the statements: (i) Df(x)=lim⁡h→0f(x)−f(x−h)hDf(x)=\lim_{h\to 0}\dfrac{f(x)-f(x-h)}{h}Df(x)=limh→0​hf(x)−f(x−h)​; (ii) Df(x)=lim⁡h→0f(x+2h)−f(x+h)2hDf(x)=\lim_{h\to 0}\dfrac{f(x+2h)-f(x+h)}{2h}Df(x)=limh→0​2hf(x+2h)−f(x+h)​. In general,

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Suppose a function f:ℜ→ℜ is differentiable at x. Consider the statements: (i) Df(x)=lim_(h→ 0)(f(x)-f(x-h))/h; (ii) Df(x)=lim_(h→ 0)(f(x+2h)-f(x+h))/2h. In general, (A) (i) is true and (ii) is false (B) (i) is false and (ii) is true (C) Both are true (D) Both are false

Also written as: Suppose a function f: Re -> Re is differentiable at x. Consider the statements: (i) Df(x)=lim_(h-> 0)(f(x)-f(x-h))/h; (ii) Df(x)=lim_(h-> 0)(f(x+2h)-f(x+h))/2h. In general,

Limits, continuity and differentiability has come up 122 times in 7 exams since 2006.

What this question tests

This question is on Differentiability and its link to continuity, part of Limits, continuity and differentiability in Mathematics.

When a derivative exists and why differentiability implies continuity but not conversely.

TestStop has 12 past-paper questions on this subtopic; the nearest in year are below.

  • IIT JAM 2021 · Q35Let f be defined on (-π/2, π/2) by f(x) = (cos (π/2 + |x|))/(sin (π/2 - |x|)). Then,
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