How do you find vertical, horizontal and oblique asymptotes for #f(x)=x(e^(1/x))#?

Question

Chloe Alexander · Accepted Answer

The oblique asymptote is #y=x+1#
The vertical asymptote is #x=0# We know the Taylor's series for #e^x=1+x+x^2/(2!)+x^3/(3!)+......# #e^(1/x)=1+1/x+1/(x^2*2!)+1/(x^3*3!+.....# #:.f(x)=xe^(1/x)=x(1+1/x+1/(x^2*2!)+1/(x^3*3!+.....))# #=x+1+1/(x*2!)+1/(x^2*3!)+...# therefore #y=x+1# is an oblique asymptote And #x=0# is a vertical asymptote y-xe^(1/x))(y-x-1)=0 [-6.815, 7.235, -2.36, 4.66]}

Christian Antunez · Answer

undefined To find the vertical, horizontal, and oblique asymptotes for $ f(x) = x(e^{1/x}) $, you would analyze the behavior of the function as $ x $ approaches different values.

1. Vertical Asymptotes: Set the denominator of any fraction in the function equal to zero and solve for $ x $. Vertical asymptotes occur at these values if the function approaches positive or negative infinity as $ x $ approaches the values where the denominator is zero.

2. Horizontal Asymptotes: Determine the behavior of the function as $ x $ approaches positive or negative infinity. If the function approaches a constant value as $ x $ becomes large in magnitude, then there is a horizontal asymptote at that constant value.

3. Oblique Asymptotes: Compute the limit of the function as $ x $ approaches positive or negative infinity. If the limit exists and is a finite constant or infinity, then there is an oblique asymptote represented by that limit value.