A Limits Question

This question involves a graph, but, unfortunately, I have no means of reproducing it. I shall try my utmost to describe it, or rather the section of it that confounds me.

Consider the function f. For what values of x_0 does the lim x-->x_0 f(x) exist, where -9<= x_0 <=4?

The answer says that at the value x_0 = -6 the limit does not exist (d.n.e.), but I think the limit is positive infinity and don't know why it d.n.e.

Here is the description:

f(-6) = 3, so there is a black point at (-6,3). Now as x approaches -6 from the left side, the curve goes upward, or without bound (+infinity). As x approaches -6 from the right side, there is another curve that goes upward, or without bound (+infinity), so it seems like there is a vertical asymptote at x_0 = -6 (but a point exists at -6).

This is the section that baffles me. If you need a clearer description, I shall try to do my best.

Thanks.
 

LeonhardEuler

Gold Member
858
1
To say that a limit is infintiy implies that the limit does not exist. The definition of a limit does not allow infinite limits (although it does allow limits where a variable is allowed to approach infinity).
 
So D.N.E. implies that the limit is not close to a single real number? I am still confused over the term . . . :confused: When is the limit in this case +infinity? Is it because of how the question is worded?? I was trying to follow that two-sided limit of a function rule. . . (Now I feel horribly lost.)

Thanks for your patience.
 
Last edited:

LeonhardEuler

Gold Member
858
1
You can write that the limit is infinity, that is not wrong, but this means that the limit does not exist. If you look at the epsilon-delta definition of the limit, you will see that a limit of infinity is impossible because it is required that the function come arbitrarily close to the limit. So, for example, if the limit is 5, the funtion must come within 1 of 5, and within .1 of 5 and any so on. But a number can not be within .1 of infinity. It's just a confusing notational thing to write limit=infinty, when this really means that limit d.n.e., and the function gets bigger than any specified value.
 

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