Limit (Calculus)

A limit of a function describes the behavior of a function near a particular input, it is one of the foundations of Calculus.

Infinite Limits

Infinite limits are limits that approach $\pm \infty$, which, as $x$ approaches creates a horizontal or slant asymptote.

If the degree of the numerator is less than the denominator the function is “bottom heavy,” meaning the limit is equal to $0$. Otherwise, if the degree of the numerator is greater than the denominator, the function is “top heavy,” meaning the limit is $\infty$.

Cases where limits fail to exist

There are a few cases where limits fail to exist:

• Limits differ from the right and left: the limit as $x$ approaches from the positive and negative side have different values. $$\begin{array}{c}{\lim }_{x\to -1^{-}}\\ {\lim }_{x\to -1^{+}}\end{array}\}\therefore \underset{x\to -1^{-}}{\lim }\ne \underset{x\to -1^{+}}{\lim }$$
• Limits display oscillating behavior: the value of the function of $x$ alternates too frequently for a limit to be discovered.
• Limits display unbounded behavior: Similar to the first case, the limit as $x$ approaches from both the negative and positive side are both equal to $\infty$. Because infinities can be of different sizes this results in the failure of being able to sufficiently check if the limits are equal. $$\begin{array}{c}{\lim }_{x\to c^{+}}=\infty \\ {\lim }_{x\to c^{-}}=\infty \end{array}\}\therefore (\underset{x\to c^{-}}{\lim }=\infty )\ne (\underset{x\to c^{+}}{\lim }=\infty )$$