Asymptotes. Find all the asymptotes of the given function, and use limits to describe them. Show all your steps

Question

f(x) = 6e^x - e^3x / e^3x - 7e^2x + 12e^x

Answer 1

You need to specify the limiting value for x!!!!

If you want the limit as x->∞, divide the numerator and denominator by e^3x to see that the limit is -1.

Answer 2

f(x) =( 6e^x - e^3x) / (e^3x - 7e^2x + 12e^x)

Let us find the domain of the function at hand.

It seems to be the set of all real numbers except the values that make the denominator zero.

e^3x - 7e^2x + 12e^x = 0

e^x { e^2x -7e^x +12 } = 0

e^{x (} e^x -3 ) ( e^x -4 ) = 0

and since e^x ≠ 0 for all real numbers we conclude that

e^x -3 =0 ⇔ x = ln3 or

e^x -4 = 0 ⇔ x = ln4

Therefore the domain of the function is ℜ - { ln3 ,ln4 } = ( -∞, ln3) ∪( ln3, ln4 )∪ ( ln4, ∞)

Now the lines x = ln3 and x = ln4 are the two vertical asymptotes of the function.

Now let us look for horizontal asymptotes

Lim _x→∞ { ( 6e^x - e^3x) / (e^3x - 7e^2x + 12e^x)}

Lim _x→∞ {e^3x ( 6e^-2x - 1) /e^3x (1 - 7e^-x+ 12e^-2x)}

Lim _x→∞ { ( 6e^-2x - 1) /(1 - 7e^-x+ 12e^-2x)} = -1 Therefore the line y = -1 is a horizontal asymptote as x→∞

Let see what happens to the function as x→-∞

Lim _x→-∞ { ( 6e^x - e^3x) / (e^3x - 7e^2x + 12e^x)} =

Lim _x→∞ {e^x ( 6 - e^2x) /e^x (e^2x - 7e^x+ 12e^x)} =

Lim _x→∞ { ( 6 - e^2x) / (e^2x - 7e^x+ 12)} = 2. That is the line y = 2 is horizontal asymptote as x→-∞.