🔦Limits

A mathematical limit tells you how to find a constant output for a function for a specified input. Depending on the function, this evaluation may fail, or may be too complicated to "see" at first.

Definition

You're given a function. The function has a constant, known output value if we evaluate the function at values very close to some chosen input value.

More precisely, given a function

$$f(x)$$

if we observe a very small changes to the output

$$\epsilon > 0$$

when given any small change to the input

$$\delta > 0$$

such that the differences in the inputs and outputs are smaller than the differences of the changes, written as

$$0< |x-a|< \delta$$

for inputs and

$$|f(x) - L| < \epsilon$$

for outputs, then we can say the limit of the function at the point of interest a is the constant L. We write this as

$$\lim \limits_{x \to a} [ f(x)] = L$$

Basic Properties

Constants may be extracted from the limit evaluation

$$\lim \limits_{x \to a }[c f(x) ] = c \lim \limits_{x \to a}[f(x)]$$

Every algebraic operation you can do on functions within a limit evaluation can be done on the evaluations of limits

Such as, addition

$$\lim \limits_{x \to a}[ f(x) + g(x)] = \lim \limits_{x \to a}[f(x)] + \lim \limits_{x \to a}[g(x)]$$

subtraction

$$\lim \limits_{x \to a}[ f(x) - g(x)] = \lim \limits_{x \to a}[f(x)] - \lim \limits_{x \to a}[g(x)]$$

multiplication

$$\lim \limits_{x \to a}[ f(x) g(x)] = \lim \limits_{x \to a}[f(x)] \lim \limits_{x \to a}[g(x)]$$

division

$$\lim \limits_{x \to a}[ \frac{ f(x) }{ g(x) }] = \frac{ \lim \limits_{x \to a}[f(x)] }{\lim \limits_{x \to a}[g(x)] }$$

exponentiations

$$\lim \limits_{x \to a}[ f(x)^n ] = \lim \limits_{x \to a}[f(x)]^n$$

and roots

$$\lim \limits_{x \to a}[ \sqrt[n]{f(x)} ] = \sqrt[n]{\lim \limits_{x \to a}[f(x)]}$$