Question

Use the graph of the function f to decide whether the value of the given quantity exists. (If an answer does not exist, enter DNE.)

(a) lim f(x)
x→ −2

(b) lim f(x)
x→0

(c) lim f(x)
x→2

(d) lim f(x)
x→4

Thanks :)

Answer 1

It's been awhile since I've studied limits, but I'll try and give it a shot. :)

Let's take a close look at the graph.

(a)

There is a dashed line at x = -2, indicating an asymptote, and the graph of f(x) appears to dive off to positive or negative infinity, depending on thedirection x approaches -2 from. This gives us two limits:

(approaching from the right)

(approaching from the left)

(b)

As x approaches 0 from the left, the value of f(x) approaches 3.5; as x approaches 0 from the right, f(x) approaches 5. Since the function iscontinuous, we have two different limits again, depending on direction:

(from theleft)

(fromthe right)

(c)

Notice that f(x) is discontinuous at 2; this leaves us unable to define a limit, as the limit of f(x) as it approaches 2 must equal f(2), which isundefined. Thus, the limit Does Not Exist.

(d)

I'm not sure about this part; here's what I think is going on:

As x approaches 4 from the left or right, the values of f(x) appear to head for infinity; the function is also defined at x = 4 as f(4) = -1; becausethe limit of the function as x approaches 4 is not equal to f(4), I don't think the limit exists.

If I'm wrong, and a limit does exist, it would be something like this:

I hoped this helped a little; sorry I'm a bit rusty on limits. Good luck! :)