Question

Determine lim x approaches 1 f(x) if f(x)=3-x, x doesn't equal to 1 and 1, x=1.

Answer 1

A good way to see this, is to graph it.

Answer 2

So for every but at the point x=1, you have the function y=3-x. If you look at that graph, tell me what you think the limit will be.

Answer 3

So the answer would be it doesn't exist, right?

Answer 4

Not quite. To find the limit, we need to find the one sided limits first.

The limit \[\lim_{x\to1^-} 3-x\]is 2. You should be able to convince yourself of this by looking at a graph of y=3-x.

Next, the limit\[\lim_{x\to1^+}3-x\]is also 2.

Answer 5

Now, since both one-sided limits equal 2, we have that \[\lim_{x\to1} f(x)=2.\]This is despite the fact that \(f(1)\neq 2\).

Answer 6

Thanks.

Answer 7

Did that all make sense to you?