For the function g whose graph is given, state the value of each quantity, if it exists. (If an answer does not exist, enter DNE.)

Question

(a)    lim t → 0⁻ g(t)

(b)    lim t → 0⁺ g(t)

(c)    lim t → 0 g(t)

(d)    lim t → 2⁻ g(t)

(e)    lim t → 2⁺ g(t)

Answer 1

When we refer to y-values, we really mean g(t).

Part (A):

limit g(t) as t -> 0- = -1

This is confirmed visually by tracing the graph from left to right and observing the y-values get closer to -1 as t approaches 0 from the left.

Part (B):

limit g(t) as t -> 0+ = -2

This is confirmed visually by tracing the graph from right to left and observing the y-values get closer to -2 as t approaches 0 from the right.

Part (C):

limit g(t) as t -> 0 = DNE

The condition for a limit to exist is that the left-hand and right-hand limits at a t-value must agree. From Parts A and B we see that this isn't the case, the left-handed limit as t approaches 0 equals -1, while the right-handed limit equals -2. Therefore the limit of g(t) as t approaches 0 does not exist.

Part (D):

limit g(t) as t -> 2- = 2

Visually determined, similar as Parts A and B.

Part (E):

limit g(t) as t -> 2+ = 0

Visually determined, similar as Parts A and B.

From Parts D and E, one would conclude that

limit g(t) as t -> 2 = DNE

Since the left-handed and right-handed limits around t = 2 differ, the limit does not exist.

Answer 2

What are the three conditions for a limit to exist?