Limit (Calculus)

A limit is the value that a mathematical function approaches to as its input approaches some value. In other words, it is the value that a function converges to, but does not necessarily equate to. Limits are used to define derivates (using first principles) and continuity, as well as to find the long-term behaviour of a function (such as with qualitative analysis).

Definition §

Limit (Calculus)

Let $f:D\to \mathbb{R}$ be a function where $D$, the domain, is a subset of all real numbers (i.e. a real-valued function). Then, the limit of $f$ as $x$ approaches $a$ is given by:

$$\underset{x\to a}{\lim }f(x)=Lx\ne a$$

Well, that’s a simplified definition. A more formal (and mathematically correct definition) can be found here: formal definition of a limit.

Limits must be:

• Unique: There cannot be two limits for a given function at the same point. This leads into the Theorem - Limits from different Sides
• Real: Limits must belong to the real set, $\mathbb{R}$
• Finite: While x can approach infinity, the limit of $f(x)$ must be finite i.e. limits can never be infinity. This property makes solving limits tricky, which is why various methods to evaluate limits exist. However,

Limit Laws §

Standard Limits §

Theorems §

T1: Limits must be equal from both sides ^t1

Let $f(x)$ be a function. For the limit,$L$ of $f$ to be defined at $x=a$, it must hold that:

$$\underset{x\to a^{+}}{\lim }f(x)=\underset{x\to a^{-}}{\lim }=L$$

• ${\lim }_{x\to a^{+}}$: $x$ approaches $a$ from the positive (right) side, getting more negative.
• ${\lim }_{x\to a^{-}}$: $x$ approaches $a$ from the negative (left) side, getting more positive.

Since limits must be unique, piecewise functions cannot have a defined limit if, as the input approaches a value, the function approaches multiple values, even if they aren’t defined at that input value. I.e. ${\lim }_{x\to a}f(x)$ cannot equal both $b$ and $c$ (or however many possible values)

Different Limits?

Take the following graph of the function:

$f(x)=\{\begin{array}{ll}1,& x>0\\ -1,& x\le 0\end{array}$

What is ${\lim }_{x\to 0}f(x)$? Is it 1 or -1?

As you can see, limits can have different values if the function is approached from the right or left side. In the example above, because ${\lim }_{x\to 0^{+}}=1$ (limit from the right side) and ${\lim }_{x\to 0^{-}}=-1$ (limit from the left side), the limits from different sides are unequal. Hence, the limit does not exist at $x=0$.

Solving Limits As X Approaches Infinity §

There are multiple methods for solving limits as x approaches infinity (since limits cannot equal infinity) that have unique conditions.

Function Form	Technique	Notes
$\frac{\infty }{\infty }$ where common terms exist	Divide numerator and denominator by the fastest growing term	The fastest growing term is also used in Big-O Notation
$\infty -\infty$	Multiply and divide by conjugate	If the function is expressed as $a+b$ then it’s conjugate is $a-b$
$\frac{\infty }{\infty }\text{ or }\frac{0}{0}$	L’Hôpital’s Rule	Both numerator and denominator functions must be differentiable
Else	Sandwich Theorem	It helps when $f(x)$ is a harmonic function, or looks to have similar upper and lower bounds.

Bivariate Limits §

See Bivariate Limit