Question

f (x) = x^2 - 4/ x - 2 if x is not equal to 2 and f (x) = 1 if x = 2. Let f be the function defined earlier. Which of the following statements about f are true? (a) f has a limit at x = 2 (b) f is continuous at x = 2 (c) f is differentiable at x = 2

Answer 1

Given:

f(x≤2)=4x+2

f(x>2)=3x+4

Function is continuous:

lim(x→2) =10 from either piece.

Each piecewise function is differentiable alone, but we see the f(x) is not differentiable because the derivatives are not the same:

(Depending on your calculus experience, you can use the definition of a derivative or the power rule to find the derivative)

For x≤2

f’=lim(h→0) {[4(x+h)+2]-[4x+2]}/h def. of derivative

f’=lim(h→0) (4x+4h+2-4x-2)/h expand and cancel terms

f’=lim(h→0) (4h/h) simplify

f’=4

or:

f’=4 power rule

For x>2

f’= lim(h→0){[3(x+h)+4]-[3x+4]}/h def. of derivative

f’=lim(h→0)(3x+3h+4-3x-4)/h expand and cancel terms

f’=lim(h→0)3h/h simplify

f’=3

or

f’=3 via power rule

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