How do you tell if a limit is going to be infinity?

[emlekarc]

How do you tell if a limit is infinity, and you should use that approach, or if you should try to factor/multiply by congegate, etc.? Do you use the latter if its 0/0 and the first if its a number/0 when you try pluging in th limit?

 

[Mark44]

There are several possibilities when you evaluate the limit expression, of which the three that you'll see most often are these.
1. The limit expression evaluates to a number. For example, $\lim_{x \to 2} \frac{x - 2}{x}$. Substituting 2 for x gives 0/2 = 0.
2. The limit expression evaluates to 0/0. For example, $\lim_{x \to 0} \frac{x^2}{x}$. The usual approaches are factoring, multiplying by the conjugate ('congegate' is not a word), L'Hopital's Rule.
3. The limit expression evaluates to some nonzero number over zero. The limit is often infinity, but you should check that you get the same sign on both the left and right sides. For example, $\lim_{x \to 0} \frac{1}{x}$. This limit doesn't exist because the left- and right-side limits aren't the same.

Item 2 above is and example of the [0/0] indeterminate form. There are several others that I haven't mentioned, including [∞/∞], [∞ - ∞], and [1^∞].