Expert answers to "How do I evaluate the limit given ?limx→∞xx−(x−1)xxx" - Plainmath

smaakt4ucq

Answered question

2022-03-24

How do I evaluate the limit given ?

lim_{x \to \infty} \frac{x^{x} - {(x - 1)}^{x}}{x^{x}}

Answer & Explanation

LiaisyAciskriuu

Beginner2022-03-25Added 10 answers

Then

lim_{x \to \infty} \frac{x^{x} - {(x - 1)}^{x}}{x^{x}} = lim_{x \to \infty} [1 - {(\frac{x - 1}{x})}^{x}] = lim_{x \to \infty} [1 - {(1 - \frac{1}{x})}^{x}]

Notice

lim_{x \to \infty} {(1 - \frac{1}{x})}^{x} = e^{lim_{x \to \infty} x \ln (1 - \frac{1}{x})}

Try making a substitution like

u = \frac{1}{x}

to get a situation in which l'Hôpital's rule applies to find this limit. Remember this will also change the value the limit approaches. It should look something like this:

e^{lim_{u \to 0} \frac{\ln (1 - u)}{u}} = e^{lim_{u \to 0} \frac{1}{u - 1}}

Jeffrey Jordon

Expert2022-07-05Added 2605 answers

Answer is given below (on video)

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