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Let f be a function so that (below). Which must be true? I. f is continuous at x=2 II. f is differentiable at x=2 III. The derivative of f is continuous at x=2 (A) I (B) II (C) I & II (D) I & III (E) II & III

#lim_(h->0)(f(2+h)-f(2))/h=5#

Answer 1

Aurora Alvarado

(C)

Noting that a function #f# is differentiable at a point #x_0# if

#lim_(h->0)(f(x_0+h)-f(x_0))/h = L#

the given information effectively is that #f# is differentiable at #2# and that #f'(2) = 5#.

Now, looking at the statements:

I: True

Differentiability of a function at a point implies its continuity at that point.

II: True

The given information matches the definition of differentiability at #x=2#.

III: False

The derivative of a function is not necessarily continuous, a classic example being #g(x) = {(x^2sin(1/x) if x!=0),(0 if x=0):}#, which is differentiable at #0#, but whose derivative has a discontinuity at #0#.

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Answer 2

Evelyn Agard

(D) I & III

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

Let f be a function so that (below). Which must be true? I. f is continuous at x=2 II. f is differentiable at x=2 III. The derivative of f is continuous at x=2 (A) I (B) II (C) I &amp; II (D) I &amp; III (E) II &amp; III

#lim_(h->0)(f(2+h)-f(2))/h=5#

Let f be a function so that (below). Which must be true? I. f is continuous at x=2 II. f is differentiable at x=2 III. The derivative of f is continuous at x=2 (A) I (B) II (C) I & II (D) I & III (E) II & III