2.1 Limits

Limit of a Continuous Function

If $ƒ\left(x\right)$ is a continuous function for all real numbers, then ${\lim }_{x\to c}ƒ\left(x\right)=ƒ\left(x\right)\left(c\right)$

Limits of Rational Functions

A. If $ƒ\left(x\right)$ is a rational function given by $ƒ\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)}$, such that $p\left(x\right)$ and $q\left(x\right)$ have no common factors, and c is a real number such that $q\left(c\right)$ = 0, then

I. ${\lim }_{x\to c}ƒ\left(x\right)$ does not exist

II. ${\lim }_{x\to c}ƒ\left(x\right)$ = $\pm \infty$ $\to$ $x=c$ is a vertical asymptote

B. If $ƒ\left(x\right)$ is a rational function given by $ƒ\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)},$ such that reducing a common factor between $p\left(x\right)$ and $q\left(x\right)$ results in the agreeable function $k\left(x\right)$, then

I. ${\lim }_{x\to c}ƒ\left(x\right)$ = ${\lim }_{x\to c}\frac{p\left(x\right)}{q\left(x\right)}=$ ${\lim }_{x\to c}k\left(x\right)$ = $k\left(c\right)$ $\to$ Hole at the point $(c,k\left(c\right))$

Limits of a Function as x Approaches Infinity

If $ƒ\left(x\right)$ is a rational function given by (x) = $\frac{p\left(x\right)}{q\left(x\right)}$, such that $p\left(x\right)$ and $q\left(x\right)$ are both polynomial functions, then

A. If the degree of $p\left(x\right)$ > $q\left(x\right)$, ${\lim }_{x\to \infty }ƒ\left(x\right)=\infty$

B. If the degree of $p\left(x\right)<q\left(x\right)$, ${\lim }_{x\to \infty }ƒ\left(x\right)=\infty$ $\to$ $y=0$ is a horizontal asymptote

C. If the degree of $p\left(x\right)=q\left(x\right)$, ${\lim }_{x\to \infty }ƒ\left(x\right)=c$, where $c$ is the ratio of the leading coefficients $\to$ $y=c$ is a horizontal asymptote

Special Trig Limits

L'Hospital's Rule

If results ${\lim }_{x\to c}ƒ\left(x\right)$ or ${\lim }_{x\to \infty }ƒ\left(x\right)$ results in an indeterminate form $(\frac{0}{0},\frac{\infty }{\infty },\infty -\infty ,0\cdot \infty ,1^{\infty },{\infty }^0)$, and $ƒ\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)}$, $\Rightarrow$

${\lim }_{x\to c}ƒ\left(x\right)=$ ${\lim }_{x\to c}\frac{p\left(x\right)}{q\left(x\right)}={\lim }_{x\to c}\frac{p^{\prime }\left(x\right)}{q^{\prime }\left(x\right)}$ and ${\lim }_{x\to \infty }ƒ\left(x\right)={\lim }_{x\to \infty }\frac{p\left(x\right)}{q\left(x\right)}={\lim }_{x\to \infty }\frac{p^{\prime }\left(x\right)}{q^{\prime }\left(x\right)}$

The Definition of Continuity

A function $ƒ\left(x\right)$ is continuous at c if:

I. ${\lim }_{x\to c}ƒ\left(x\right)$ exists

II. $ƒ\left(c\right)$ exists

III. ${\lim }_{x\to c}ƒ\left(x\right)=ƒ\left(c\right)$

Types of Discontinuities

Removable Discontinuities (Holes):

I. ${\lim }_{x\to c}ƒ\left(x\right)=L$ (the limit exists)

II. $ƒ\left(c\right)$ is undefined

Non-Removable Discontinuities (Jumps and Asymptotes):

A. Jumps:

${\lim }_{x\to c}ƒ\left(x\right)$ = DNE because ${\lim }_{x\to c^{-}}ƒ\left(x\right)$ $\ne$ ${\lim }_{x\to c^{+}}ƒ\left(x\right)$

B. Asymptotes (Infinite DIscontinuities):

${\lim }_{x\to c}ƒ\left(x\right)=\pm \infty$

Intermediate Value Theorem

If $ƒ$ is a continuous function on the closed interval $[a,b]$ and $k$ is any number between $ƒ\left(a\right)$ and $ƒ\left(b\right)$, then there exists at least one value of $c$ on $[a,b]$ such that $ƒ\left(C\right)=k$ . In other words, on a continuous function, if $ƒ\left(a\right)<ƒ\left(b\right)$, any $y$ value greater than $ƒ\left(a\right)$ and less than $ƒ\left(b\right)$ is guaranteed to exist on the function $ƒ$.

Average Rate of Change

The average rate of change, $m$, of a function $ƒ$ on the interval $[a,b]$ is given by the slope of the secant line.

$m=\frac{ƒ\left(b\right)-ƒ\left(a\right)}{b-a}$

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