# How do limits work in calculus?

Jul 27, 2015

The same way they work in Pre-Calculus.

Limits convey the action of approaching a coordinate in a graph that may or may not exist in the curve itself, whether it's due to an asymptote or a discontinuity. It tends to describe a value that you are unsure exists and can be used as a systematic way of determining whether or not it does. You can approach from the left or right in a $y = f \left(x\right)$ graph.

For example:
${\lim}_{x \to {0}^{+}} \frac{1}{x} = \infty$
"The limit as x approaches 0 from the positive/right side of $\frac{1}{x}$ is infinity"

${\lim}_{x \to {0}^{-}} \frac{1}{x} = - \infty$
"The limit as x approaches 0 from the negative/left side of $\frac{1}{x}$ is negative infinity"

${\lim}_{h \to 0} \frac{f \left(x + h\right) - f \left(x\right)}{h}$
"The limit as $h$ approaches a very small number from either side of $\frac{f \left(x + h\right) - f \left(x\right)}{h}$ is the slope of the function $h$ units away from the point of examination"

which basically says that if you zoom in very far into a function, it looks linear and you can use the basic slope formula at that close zoom to find the slope. $h$ is some very small value, hence the $h \to 0$.