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

Question

kinggeorge · Answer

A good way to see this, is to graph it. |dw:1353365544506:dw|

kinggeorge · Answer

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.

anonymous · Answer

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

kinggeorge · Answer

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.

kinggeorge · Answer

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

anonymous · Answer

Thanks.

kinggeorge · Answer

Did that all make sense to you?