Find the limit of the function.

Question

Find the limit
lim   (5(e^−x) + 6)/(6(e^−x) + 5) 
x→∞

For questions like this I was taught to look at the starting coefficients, but I don't know how to do that when e is involved. I thought maybe the e's and their factors cancel and it was 5/6 but that was wrong. Could you please show me how to do this problem?

Thank you.

Answer 1

lim_x→∞[(5e^-x+ 6)/(6e^-x+5)] = lim_x→∞[(5/e^x+6)/(6/e^x+5)]

 

                                          = (0+6)/(0+5)

 

                                          = 6/5

Answer 2

If you plug in infinity into the function, you get an indeterminate, which is ∞/∞.  So you must perform L'Hospital Rule.  You take the derivative of the numerator over the derivative of the denominator.

 

lim         -5e^-x

x->∞   ________

              -6e^-x

 

 

Then notice that e^-x cancel each other out. 

 

lim       -5

x->∞  ____

             -6

 

 

Since the function is now a constant, the limit as x approaches infinity is 5/6.