Limit

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

Definition & Properties §

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

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

• As x approaches a, f(x) approaches (or converges) to L

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

Theorem - Limits from different Sides §

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. Hence, to solve this problem a simple theorem exists:

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

In plain words, the limit exists at a point a if, and only if, the right hand and left hand side limits both exist and are equal. 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 $x=0$.

Limit Laws §

Standard Limits §

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
$f(x)=\frac{\infty }{\infty }$ where common terms exist	Divide numerator and denominator by the fastest growing term.	The fastest growing term is also used in Big-O notation
$f(x)=\infty -\infty$	Multiply and divide by conjugate	If the function is expressed as $a+b$ then it’s conjugate is $a-b$
$f(x)=\frac{\infty }{\infty }$ or $\frac{0}{0}$	L’Hôpital’s Rule	Both numerator and denominator functions must be differentiable
None of the previous methods work	Sandwich Theorem	It helps when $f(x)$ is a harmonic function, or looks to have similar upper and lower bounds.

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• Continuity
• Formal Definition Of A Limit
• L’Hôpital’s Rule
• Limit Laws
• Limits and Continuity of Bivariate Functions
• Sandwich Theorem
• Standard Limits

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