Definitions

Precise Definition

We say that the limit of $f(x)$ as $x$ approaches $a$ is equal to $L$ if for every $\epsilon > 0$, there exists a $\delta > 0$ such that whenever $0 < |x - a| < \delta$, then $|f(x) - L| < \epsilon$.

"Working" Definition

The limit of $f(x)$ as $x$ approaches $a$ equals $L$ if $f(x)$ can be made as close to $L$ as we want by choosing $x$ sufficiently close to $a$, from either side, without letting $x$ equal $a$.

Right Hand Limit

$\lim_{x \to a^+} f(x) = L$ implies that as $x$ approaches $a$ from the right, the value of $f(x)$ approaches $L$.

Left Hand Limit

$\lim_{x \to a^-} f(x) = L$ implies that as $x$ approaches $a$ from the left, the value of $f(x)$ approaches $L$.

Limit at Infinity

$\lim_{x \to \infty} f(x) = L$ if $f(x)$ can be made as close to $L$ as we want by taking $x$ to be sufficiently large and positive.

Similarly, $\lim_{x \to -\infty} f(x) = L$ if $f(x)$ can be made as close to $L$ as we want by taking $x$ to be sufficiently large and negative.

Infinite Limit

$\lim_{x \to a} f(x) = \infty$ if we can make $f(x)$ arbitrarily large (positive) by taking $x$ sufficiently close to $a$ without letting $x$ equal $a$.

Similarly, $\lim_{x \to a} f(x) = -\infty$ if we can make $f(x)$ arbitrarily large and negative.

References

• calculus_cheat_sheet_limits | tutorial.math.lamar.edu