I need help on calc 2

Question

find limits

 

lim

x→∞  e^x/x^3

Answer 1

lim (e^x / x³) as x→∞ is an indeterminate form, so use L'Hopital's rule as many times as you need to get a determinate form.

 

e^x / x³  →  e^x / (3x²)  →  e^x / (6x)  →  e^x / 6

 

lim (e^x / x³) = lim (e^x / 6) = ∞

x→∞              x→∞

Answer 2

lim_(x→∞)[e^x/x³] has the form ∞/∞

 

= lim_(x→∞)[e^x/(3x²)]  (By L'Hopital's Rule)

 

      The limit still has the form ∞/∞, so apply L'Hopital's Rule again.

 

= lim_(x→∞)[e^x/(6x)]

 

      The limit again has the form ∞/∞, so apply L'Hopital's Rule once          more.

 

= lim_(x→0)(e^x/6) = ∞

 

 

Answer 3

The vertical asymptote of this function is x=0.  Since we want to know what happens to the function values as x increases infinity, we choose value of x that are greater than zero and beyond.

 

Evaluate the function values in the order given:

 

f(1)

f(10)

f(100)

f(10000)

f(1000000)

 

What happens to the function values as x goes from  x=1 , x=10, x=100, x=10000, and x=1000000 in this order?  Do they get closer to a certain y-coordinate, or do they increase infinity?  Decrease infinity?