Home > Calculus > Determining Limits Algebraically

What is the limit of #((e^x)-x)^(2/x)# as x approaches infinity?

Answer 1

#e^2#

#(e^x-x)^(2/x)=e^2(1-x/e^x)^(2/x)#

by the binomial expansion

#(1-x/e^x)^(2/x)=1+2/x(-x/e^x)+2/x(2/x-1)(-x/e^x)^2 1/(2!)+cdots# #=1-2/e^x+(p_2(x))/e^(2x)+cdots+(-1)^n(p_n(x))/e^(nx)1/(n!)+cdots# where #p_n(x)# is a #n# degree polynomial.

We know that #lim_(x->oo)(p_n(x))/e^x = 0# so

#lim_(x->oo)(e^x-x)^(2/x) = e^2#

By signing up, you agree to our Terms of Service and Privacy Policy

Answer 2

Emily Ammerman

The limit of ((e^x)-x)^(2/x) as x approaches infinity is e^2.

By signing up, you agree to our Terms of Service and Privacy Policy

Answer 3

Scarlett Allison

The limit of ((e^x - x)^{\frac{2}{x}}) as (x) approaches infinity is (e^2).

By signing up, you agree to our Terms of Service and Privacy Policy

Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.