find the limit.
lim (5-e^x)/(5+3e^x)
(x-> infinity)
how would i go about solving this?
thank you in advance to anyone who can help me.
im pretty sure my first step would be to divide the numerator and denominator by e^x but i don't know where to go after that.
Good idea, then your expression would be
(5/e^x - 1)/(5/e^x + 3)
Now, as x ---> infinity , both of the 5/e^x ----> 0
and you are left with
(0-1)/(0+3) = -1/3
(test with your calculator with x = 100)
oh okay i understand now! thank you so much for helping me. i really do appreciate it!
To find the limit of the given function as x approaches infinity, you can use the concept of limits and algebraic manipulations. Here's the step-by-step process:
1. Begin by focusing on the terms with the highest power of e, which is e^x in this case. Divide both the numerator and denominator by e^x to simplify the expression:
lim (5 - e^x)/(5 + 3e^x)
= lim (5/e^x - 1)/(5/e^x + 3)
2. As x approaches infinity, e^x also approaches infinity. So, when x is very large, both 5/e^x and 3/e^x become very close to zero. Therefore, we can make the assumption that they are negligible compared to 1. Rewriting the expression:
lim (5/e^x - 1)/(5/e^x + 3)
= lim (0 - 1)/(0 + 3)
3. Now, evaluate the limit:
lim (0 - 1)/(0 + 3)
= lim (-1)/(3)
= -1/3
Therefore, the limit of (5 - e^x)/(5 + 3e^x) as x approaches infinity is -1/3.
Note: In step 2, we used the principle that when dealing with the limit of a ratio of functions, if the numerator and denominator approach zero individually, or if the numerator and denominator approach infinity individually, we can compare their magnitudes.