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BAS CAL | Limits

Limit of a Function

the behavior of a function at a certain point
represented as f(x) → ?
- the limit of the function, f(x), as x approaches ?

Limit Equation

\lim_{x\to a}f\left(x\right)=L

where:

L is the limit of f(x) as x approaches a

Limits vs Values

f(a) is the value of the function at x=a
\lim_{x\to a}f\left(x\right)=L is the value as x approaches but not equal to a.
The limit might not always exist.
The limit is usually equal to f(a), but that is not always the case.

Existence of Limits

The limit does not exist when:
- The left hand limit (a^-) and right hand limit (a⁺) are not equal
  - “d.n.e.” or “does not exist” must be written
- The function increases or decreases without bounds in either direction
  - “-∞” or “+∞” must be written
- the value increases without bounds from one side, and decreases without bounds from the other
  - “d.n.e.” or “does not exist” must be written

One-sided Limits

used if:
- the function’s domain only uses one side of the number
- the function behaves differently on both sides of a number
common in radical and piecewise functions, as well as ones involving absolute value

Right-hand Limit

\lim_{x\to a^{+}}f\left(x\right) | applies when you are only using numbers greater than a

Left-hand Limit

\lim_{x\to a^{-}}f\left(x\right) | applies when you are only using numbers less than a

Informal Way of Solving for Limits

Table of Values

Using the given function, make two tables of graphs where the x-values are getting increasingly close to or approaching a. One should have the values increase, and the other should have the value decrease.
Write their corresponding y-values
Never use a in the table of graphs.
Based on the y-values, infer the limit

Graph

Use a graph to interpret the limit provided.
Just because there is a hole at a does not mean the y-coordinate of the hole is not the limit.
The limit only exists if the height being approached is the same for the left-hand and right-hand limits.
If the limit increases/decreases without bounds, then it is infinite.

Limit Laws

\lim_{x\to a}c=c | The limit of a constant is the given constant.
\lim_{x\to a}x=a | The limit of of the identity function is a.
\lim_{x\to a}\left\lbrack cf\left(x\right)\right\rbrack=c\lim_{x\to a}f\left(x\right) | The limit of a constant multiple

\lim_{x\to a}\left\lbrack f\left(x\right)\right.+g\left(x\right)]=\lim_{x\to a}f\left(x\right)+\lim_{x\to a}g\left(x\right) | The limit of a sum is the sum of their limits.
\lim_{x\to a}\left\lbrack f\left(x\right)\right.-g\left(x\right)]=\lim_{x\to a}f\left(x\right)-\lim_{x\to a}g\left(x\right) | The limit of a sum is the sum of their limits.
\lim_{x\to a}\left\lbrack f\left(x\right)\right.g\left(x\right)]=\lim_{x\to a}f\left(x\right)\cdot\lim_{x\to a}g\left(x\right) | The limit of a product is the product of their limits.
\lim_{x\to a}\left\lbrack\frac{f\left(x\right)}{g\left(x\right)}\right\rbrack=\frac{\lim_{x\to a}f\left(x\right)}{\lim_{x\to a}g\left(x\right)} | The limit of a quotient is the quotient of their limits.
\lim_{x\to a}\left\lbrack f\left(x\right)\right\rbrack^{n}=\left\lbrack\lim_{x\to a}f\left(x\right)\right\rbrack^{n} | The limit of the nth power of a function is the nth power of its limit.
\lim_{x\to a}\sqrt[n]{f\left(x\right)}=\sqrt[n]{\lim_{x\to a}f\left(x\right)} | The limit of the nth root of a function is the nth root of its limit.

Substitution Principle

As long as f(a) is a real number or well-defined:

\lim_{x\to a}f\left(x\right)=f\left(a\right)

Indeterminate Forms

used when f\left(a\right)=\frac00

Factoring

Factor the numerator/denominator and cancel out the common factors so that the denominator is no longer equal to 0.
Find the limit as usual.

Rationalizing

Find the conjugate of the numerator or denominator, depending on what you need to rationalize.
Multiply the numerator and denominator by the conjugate.
If necessary, cancel the terms before solving.

Infinite Limits

occurs when the function value increases/decreases without bounds

Vertical Asymptote

In rational functions, infinite limits occur when a is substituted for x and the denominator is equal to 0.
- This entails that there is a vertical asymptote at x=a

Rules

Let a, k be real numbers and k≠0. In \lim_{x\to a}\frac{f\left(x\right)}{g\left(x\right)} , \lim_{x\to a}f\left(x\right)=k and \lim_{x\to a}g\left(x\right)=0

If k > 0 and g(x) → 0 through positive values, then \lim_{x\to a^{+}}\frac{f\left(x\right)}{g\left(x\right)}=+\infty
If k > 0 and g(x) → 0 through negative values, then \lim_{x\to a^{-}}\frac{f\left(x\right)}{g\left(x\right)}=-\infty
If k < 0 and g(x) → 0 through positive values, then \lim_{x\to a^{+}}\frac{f\left(x\right)}{g\left(x\right)}=-\infty
If k < 0 and g(x) → 0 through negative values, then \lim_{x\to a^{-}}\frac{f\left(x\right)}{g\left(x\right)}=+\infty
Note: If the whole expression in the denominator is raised to an even power, the sign of the limit will match the one of the numerator.

Limits at Infinity

Rational Functions

\lim_{x\to+\infty}\frac{c}{x^{n}}=0
\lim_{x\to-\infty}\frac{c}{x^{n}}=0

Using these concepts, you can do the following to solve for the limit of a rational function:
1. Divide each term in both the numerator and the denominator by the highest power of x in the denominator.
2. Get the limit of each term based on the theorems above and the limit of a constant rule.
In simpler words:
- If the largest power is in the denominator, then the limit is zero.
- If the largest power is in the numerator, then the limit is the quotient of the numerator’s leading coefficient and the denominator’s leading coefficient.