`lim_(x->oo) (x - ln(x))` Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply,...

`lim_(x->oo) (x - ln(x))` Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.

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Textbook Question

Chapter 4, 4.4 - Problem 53 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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gsarora17 | (Level 2) Associate Educator

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`lim_(x->oo)x-ln(x)`

`=lim_(x->oo)x(1-ln(x)/x)`

`=lim_(x->oo)x(1-lim_(x->oo)ln(x)/x)`

Now let us evaluate

`lim_(x->oo)ln(x)/x`

Apply L'Hospital rule , Test condition:

`lim_(x->oo)((ln(x))')/(x')=1/x=1/oo=0`

`=lim_(x->oo)x(1-0)`

plug in the value,

`=oo(1-0)=oo`

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