SOLUTION: Dear Sir/Madam, I am confronted with the following problem: "Find an exponential function of the form f(x) = a * e^(b/x) + c that has y-intercept (0,5), a horizontal asymptote

Question 6725: Dear Sir/Madam,
I am confronted with the following problem:
"Find an exponential function of the form f(x) = a * e^(b/x) + c that has y-intercept (0,5), a horizontal asymptote y = 3, and whose graph passes through the point (4,10)."
I don't know how to use the fact that the function has a horizontal asymptote at y = 3 in order to solve this question. I know that the horizontal asymptote means that the function will never be equal to 3, but other than that, I don't know how to use it. Can you help me?
Thanks in advance.
Regards,
-Mike
Answer by khwang(438) (Show Source): You can put this solution on YOUR website!
Find an exponential function of the form f(x) = a * e^(b/x) + c that has y-intercept (0,5), a horizontal asymptote y = 3, and whose graph passes through the point (4,10)."

"a horizontal asymptote y = 3" means " when x-->+/- oo, y-->3"
(you are right my can never be 3.]
In this exponential case, if b> 0 and lim f(x) exists as x-->+oo,
then lim f(x)--> a*0 + c= c, so y=c=3.
When b > 0, lim f(x) --> c=3 as x-->-oo.
We claim, c = 3.
Now, f(0) = a*1 + 3 = 5, so a = 2.

Next, use f(4) = 2e^(b/4) + 3 = 10, or e^(b/4) = 7/2.
So b = 4ln(7/2) [stop here, don't make ugly decimals or approx.]
Answer f(x) = 2 e^(4/x * ln(7/2)) +3.
[Note ; e^(4/xln(7/2)) = (7/2)^(4/x)]
Kenny
PS: It is important for me that you posted these two questions in
wrong category. Never happen again!

RELATED QUESTIONS
Dear Sir/Madam, I am confronted with the following problem: "Find an exponential... (answered by longjonsilver)

Dear Sir/Madam, I am confronted with the following problem: "Where does the graph... (answered by ichudov,dval9159)

Dear Sir/Madam, I am confronted with the following problem: "Solve the... (answered by rapaljer)

Dear Sir/Madam, I am confronted with the following problem: "The angles with... (answered by longjonsilver)

Dear Sir/Madam, I am confronted with the following problem: "The expression... (answered by longjonsilver)

Dear Sir/Madam, I am confronted with the following problem: "Evaluate {{{ log( 5,... (answered by ichudov)

Dear Sir/Madam, I am confronted with the following problem: "Find the center and... (answered by rapaljer)

Dear Sir/Madam, I am confronted with the following problem: "Rationalize the... (answered by rapaljer)

Dear Sir/Madam, I am confronted with the following problem: "Let parallelogram ABCD... (answered by prince_abubu)