Limits, Which of the following are true statements about this function?

Question

Let f(x) = 3x^2 - 5, x≤ 1  AND 6x+2, x>1

Which of the following are true statements about this function? Which are false?

I. lim x→1 f(x) exists

II. lim x→1 f'(x) exists

III. f'(1) exists

Answer 1

Lim_x→1-  f(x) = 3(1)²-5 = -2 and lim_x→1⁺  f(x) = 6(1)+2 = 8

 

Since the one-sided limits are unequal, I is FALSE.

 

Since I is false, the function is not continuous at x = 1.  Therefore, the function is not differentiable at x = 1 either.  So, III is also FALSE.

 

lim_x→1^-  f'(x) = 6(1) = 6 and lim_x→1⁺  f'(x) = 6.

 

Since the one-sided limits are the same, lim_x→1f'(x) = 6

 

So, II is TRUE.