asymptotes??

Question

i need to find the equations of all lines of asymptote for these equations but i don't know how
can someone please check my answers for these?

a. y = 3* (2^x) - 1 y = -1

b. y = 2 / (3+x) x = -3

c. y = (2x-1) / (x) x = 0

d. y = (3x+1000) / (x-5) x = 5

e. y = 2* [(3/4)^x] + 2 y = 2

oops, its supposed to read
a. y = 3* (2^x) - 1
y = -1
and so on

Answer 1

An asymptote is a value that a function approaches but never reaches.

a. as x-> -Infinity 2^x -> 0 and y -> -1

b. y = 2 / (3 + x)
Now (3 + x) -> 0 as x -> -3 and this results in divide by zero
So x = -3 is an asymptote

Also consider when x goes to +/- infinity
As this happens y -> 0 When x is < -3, y approaches 0 from the negative side and when x > -3 it approaches from the positive side
So this is another asymptote

c. (2x - 1) / (x) Has an asymptote at x= 0 since you can not divide by 0
Rewrite so: y = (2 - 1/x)/1 by dividing top and bottom by x
Now as x-> +/- Infinity 1/x -> 0
So y = 2 is also an asymptote

d.y = (3x+1000) / (x-5)
If x= 5 then x-5 = 0 and you can not divide by 0
So this is an asymptote

Also consider when x goes to +/- infinity
rewrite the equation: (3 + 1000/x) / (1 - 5/x) by dividing top and bottom by x
Now as x-> +/- infinity 1000/x and 5/x got to 0 leaving y = 3
This is another asymptote

e. y = 2* [(3/4)^x] + 2
As x-> +Infintiy (3/4)^x ->0 and y = 2
As x-> -Infinity (3/4)^x -> +Infinity
This is the only asymptote

Answer 2

Good job! You can find two more asymptotes.
c. y = 2, horizontal asymptote
d. y = 3, horizontal asymptote