Find the limit. Use l'Hospital's Rule if appropriate. If there is a more elementary method, consider using it. lim x?8 (1+2/x)^x

Question

I need detailed solution with working shown

Answer 1

Do you mean x→∞ or x→8.

 

The latter is easy as you can just plug in x = 8 to get (1+2/8)⁸ = 390625/65536

 

More likely though, you mean the former. In that case you will need LHopital's rule.

 

(1 + 2/x)^x = exp(x ln(1+2/x)) = exp(ln(1+2/x) / x^-1)

 

lim_x→∞ ln(1+2/x) / x^-1

 

= lim_x→∞ (1 + 2/x)^-1(-2x^-2)/(-x^-2)

 

= lim_x→∞ 2/(1+2/x)

 

= lim_x→∞ 2x/(x+2)

 

= 2

 

Thus the original limit is

 

lim_x→∞ (1 + 2/x)^x = e²