what is $\lim_{x\to6}\frac{5}{(x-6)^2}$ $$\lim_{x\to6}\frac{5}{(x-6)^2}$$
Is it undefined or infinity and why ?
 A: If $x$ is near but not equal to $6$, then $(x-6)^2$ is a small positive number, and $\frac5{(x-6)^2}$ is therefore a large positive number. The closer $x$ gets to $6$, the larger this number gets, and there is no upper bound on how large it can be. Thus, we write
$$\lim_{x\to 6}\frac5{(x-6)^2}=+\infty\;.\tag{1}$$
Since $+\infty$ is not a real number, the limit in question does not exist as a real number; in the set $\Bbb R$ of real numbers the limit does not exist. However, it fails to exist in a very particular way: the function $f(x)=\frac5{(x-6)^2}$ ‘explodes’ upwards as $x$ gets closer and closer to $6$. The notation $(1)$ is an abbreviated statement of this fact, and in words we often say simply that the limit is $+\infty$, understanding that this is not a real number.
At a more advanced level one deals with the extended real numbers, which do include $-\infty$ and $+\infty$, and in that context the limit in $(1)$ actually does exist. That, however, is not the usual first-year calculus context.
