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:

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

• 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:

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

Limit Laws

Just as most operators, like the integral and $\frac{d}{dx}$ have their own respective laws as to what addition, multiplication, etc. mean for them, limits also have certain limit laws. These help decompose a complex limit into a simpler version.

Laws §

1. ${\lim }_{x\to a}[f(x)+g(x)]={\lim }_{x\to a}f(x)+{\lim }_{x\to a}g(x)$

Addition does not affect limits, we can ‘pull out’ the plus sign from the limit. E.g. ${\lim }_{x\to \infty }[x^2+\log (x)]={\lim }_{x\to \infty }(x^2)+{\lim }_{x\to \infty }(\log (x))$

2. ${\lim }_{x\to a}[c\times f(x)]=c\times {\lim }_{x\to a}f(x)$

Constant factors multiplication (and division) do not affect limits. E.g. ${\lim }_{x\to \infty }\left[\frac{39}{x^2}\right]=39\times {\lim }_{x\to \infty }\left(\frac{1}{x^2}\right)$

3. ${\lim }_{x\to a}[f(x)\times g(x)]={\lim }_{x\to a}f(x)\times {\lim }_{x\to a}g(x)$

4. ${\lim }_{x\to a}\left[\frac{f(x)}{g(x)}\right]=\frac{{\lim }_{x\to a}f(x)}{{\lim }_{x\to a}g(x)}if{\lim }_{x\to a}g(x)\ne 0$

5. ${\lim }_{x\to a}c=cifc\ne \infty$

6. ${\lim }_{x\to a}x=aifa\ne \infty$

Link to source

Standard Limits §

Standard Limits

The following limits do need any further proof to be used. Make sure to say that they are standard limits in working out.

1. ${\lim }_{x\to \infty }\frac{1}{x^p}=0,(p>0)$

2. ${\lim }_{x\to \infty }{r}^x=0,(|r|<1)$

3. ${\lim }_{x\to \infty }{a}^{\frac{1}{x}}=1,(a>0)$

4. ${\lim }_{x\to \infty }{x}^{\frac{1}{x}}=1$

5. ${\lim }_{x\to \infty }\frac{a^x}{x!}=0,(a\in \mathbb{R})$

6. ${\lim }_{x\to \infty }\frac{\log (x)}{x^p},(p>0)$

7. ${\lim }_{x\to \infty }(1+\frac{a}{n}{)}^n=e^a,(a\in \mathbb{R})$

8. ${\lim }_{x\to \infty }\frac{n^p}{a^n}=0,(p\in \mathbb{R},a>1)$

Link to source

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 [[Complexity Analysis#asymptotic-notation	fastest growing term]].
$f(x)=\infty -\infty$	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	L’Hôpital’s Rule]]
None of the previous methods work	[[Sandwich Theorem	Sandwich Theorem]]