Evaluate the limit. (I could not figure out the answer)

Question

lim x → ∞ (xe^6/x-x)

Answer 1

lim_(x→∞) [x(e^6/x - 1)] = lim_(x→∞) [ (e^6/x - 1) / (1/x)], which has the indeterminate form 0/0.

So, applying L'Hopital's Rule, the given limit is equivalent to:

lim_(x→∞) [(e^6/x(-6/x²) / (-1/x²)] = lim_x→∞) [6e^6/x] = 6e⁰ = 6

Answer 2

Note that e^x = x⁰/0! + x¹/1! + x²/2! + ... = ∑xⁿ/n! over n = 0..inf

So, e^6/x = (6/x)⁰/0! + (6/x)¹/1! + (6/x)²/2! + ...

Then, xe^6/x - x = x(e^6/x - 1) = x((6/x)⁰/0! + (6/x)¹/1! + (6/x)²/2! + ... - 1) = x((6/x)¹/1! + (6/x)²/2! + ...) = 6¹/x⁰1! + 6²/x¹2! + ...

Therefore lim(x→inf) xe^6/x - x = 6