calculus functions

Question

Sketch the graph of the function.

f(x) = 4 + x if x < −1

x² if −1 ≤ x < 1

2 − x if x ≥ 1

Use the graph to determine the values of a for which lim x → a f(x)

 does not exist. (Enter your answers as a comma-separated list.)

Answer 1

Brigitte

The limit will not exist at x=a if the slope of the tangent line is different coming from the negative and the positive direction to x=a.

So at x = -1. Coming from the left we come along f(x) = 4+x. f'(x) = 1. So at x=-1 slope of tangent line from the left is 1. Coming from the right we come along f(x) = x². f'(x) = 2x, so when x=1 slope of tangent line from the right is -2. 1 and -2 are not the same so the limit does not exist when x= -1.

Next consider x = 1.

From the left we come along f(x)=x². f'(x) = 2x, so slope of tangent line from left is 2.

From the right we come dalong f(x) = 2-x. f'(x) = -1. So slope of tangent line from right is -1.

2 and -1 are different, so the limit does not exist at x = 1.

4+x, x², 2-x are all differentiable at all points, so the only possible x values where f(x) can not be differentiable (there the limit doesnt exist) are at the points of intersection ie at x = -1, 1.

Mike