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.)

Question

4+x if x<-1

f(x)={ 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.)

a=

Answer 1

Hey Simon!

This question relies on the idea that, for a limit to exist at a point a, the right and left-hand limits must be equivalent at a.

To find out what the function equals as x approaches -1 from the left, we use the first equation in your Piecewise function, y = 4 + x.

4 + (-1) = 3

As x approaches -1 from the right, we use y = x². (-1)² = 1.

Because the left and right hand limits are different at -1, the limit does not exist at x = -1.

Now, we use the same steps to check x = 1, using the equations y = x² from the left and y = 2 - x from the right. 1² = 1 and 2 - 1 = 1. In both cases, y = 1. Because the y values are the same at the point x = 1, the limit exists.

The only point at which the function does not exist is x = -1.

The question asks you to use the graph to answer the question. While we used the equations to find the answer above, you can use the graph by looking for "jumps."

Answer 2

Have you tried looking at the graph? In order for the limit as x->a to exist, the left and right hand limits must both exist and be equal.