<aside> ⛔ Given a function $f(x)$ and a fixed point $a$, we say that $f$ approaches the limit $l$ near $a$, provided we can make $f(x)$ ****as close to $l$ as we like by taking $x$ sufficiently close to, but not equal to, $a$, and write:

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• If we cannot make $f(x)$ as close to $l$ as we like for any $l$ by taking $x$ sufficiently close to, but not equal to, $a$, then we say that $f$ does not have a limit as $x$ approaches $a$; and write:

Conditions for a limit to exist:

• One-sided limits must be equal

One-Sided Limits

Quotient Law

$lim_{x \to a} f(x) = L$ and $lim_{x \to a} g(x) = M$, $lim_{x \to a} \frac{f(x)}{g(x)} = \frac{L}{M}$

Sum Law

$lim_{x \to a} f(x) + g(x) = L+M$

Product Law

$lim_{x \to a} f(x) = L$ and $lim_{x \to a} g(x) = M$, then $lim_{x \to a} f(x) g(x) = LM$

Infinite Limits

Limits towards vertical asymptotes

$lim_{x \to a^+} \frac{1}{x-a}$ does not exist, with $lim_{x \to a^+} \frac{1}{x-a} \to + \infty$

Similarily,

$lim_{x \to a^-} \frac{1}{x-a}$ does not exist, with $lim_{x \to a^-} \frac{1}{x-a} \to - \infty$