Calculus Unit 1 Asymptotes

Horizontal Asymptotes

100

What is Horizontal Asymptotes 

The horizontal asymptote of a function is a horizontal line to which the graph of the function appears to coincide with but it doesn't actually coincide.

200

What are the rules of horizontal asymptotes?

• If n < m, the horizontal asymptote is y = 0.
• If n = m, the horizontal asymptote is y = a/b.
• If n > m, there is no horizontal asymptote.

300

What is the horizontal asymptote of the function?

f(x)=((x+3)(x+2))/((x+5)(x-1))

y=1

(same power - divide the coefficients 1/1=1)

400

What is the horizontal asymptote of the function?

f(x)=((3x^2+2)(x+4))/((x^2+4)(x+1)）

y=3

(same power - divide the coefficients 3/1=3)

500

What is the horizontal asymptote of the function?

f(x)=(x(x+5))/(x+1)

y=x+4

(numerator has a higher power - long division)

Vertical Asymptotes

100

How do we find vertical asymptotes? 

set the denominator equal to 0 and solve for x

200

How many vertical asymptotes does the function have?

f(x)=(x^2-7x+12)/x

0

300

How many vertical asymptotes does the function have?

f(x)=(x^2-8x+16)/(x-4)=((x-4)(x-4))/(x-4)

x=4

400

What are the vertical asymptote(s) of the function?

f(x)=(x+2)/(x^2-x-6)

x=3

500

What are the vertical asymptote(s) of the function?

f(x)=(x^2+10x+25)/(x^2+3x-10)

x=2

Limits

100

limx->oo

f(x)=(x-1)/(x^2+x-6)=(x-1)/((x-2)(x+3)

limx->oo=0

200

Graph the function and find 

limx->-oo

f(x)=(x-1)/(x^2+5x+6)=(x-1)/((x-2)(x+3)

limx->oo=0

300

Graph the function and find 

limx->2^-

f(x)=(x^2+1)/(x+2)

limx->2^(-)=oo

400

Graph the function and find 

limx->-4^+

f(x)=(2x^2+1)/(x^2+16)

limx->2^(+)=-oo

500

Graph the function and find 

limx->∞

f(x)=((x+3)(x+2)(2x))/((x+5)(x-1))

∞ 

no horizontal asymptote

Limit Trig

100

lim x->∞ sin (6x)

DNE

200

lim x-> ∞ (sin x)/x

y=o

300

lim x->∞ 

f(x)=(sinx)/x

0

400

lim x->∞ 

f(x)=sin(1/x)-((6x^2+2x)/(3x^2))

-2

500

lim x->∞ 

f(x)=cos((2x-pix^2)/(x^2))

y=-1