Why Do Limits Differ When Approaching from Different Sides?

[Asphyxiated]

Homework Statement

I just have a question about these two questions:

$$\lim_{y \to 3^{+}} \frac {y+1}{(y-2)(y-3)}$$

and

$$\lim_{y \to 3} \frac {y+1}{(y-2)(y-3)}$$

the solution to the first problem is infinity and the solution to the second is does not exist, is this because the first one is is approach 3 from the positive side so its basically 3.000000000000...1 and therefore infinity and the second is at 3 so the denominator is 0? If i am wrong please clarify this issue for me.

thanks!

 

[physicsman2]

The first one is infinity because you're essentially dividing by a very small positive number since you're approaching 3 from the right. When applying the limit, you will get something like 4/(1 x 0^+), meaning your dividing 4 by a very very small number therefore the limit is infinity.

The second one does not exist because if you take the one sided limits, you will get +infinity when you approach 3 from the right and -infinity when you approach 3 from the left. Since the limits from both sides don't match, the limit as x approaches 3 DNE.