Question

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?

lim x→[infinity] (4x − ln(x))

Answer 1

4

Explanation:

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?

Given the limit of the function

lim x→[infinity] (4x − ln(x))

Step 1: Substitute x = ∞ into the given function first as shown;

lim x→[infinity] (4x − ln(x))

= (4(∞) − ln(∞))

= ∞ - ∞ (indeterminate)

Step 2: Apply l’Hospital’s Rule

`\lim_(n \to \infty) (4x - lnx)\\ \lim_(n \to \infty) (d)/(dx)(4x-lnx) \\ \lim_(n \to \infty) (4 - (1)/(x)\\`

Step 3: Substitute the value of x into the resulting function;

`\\lim_(n \to \infty) (4 - (1)/(x))\\= 4 - 1/ \infty\\= 4 - 0\\= 4`

Hence the limit of the function is 4. And yes, the L'hospital rule was applicable.