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] √ x /lnx

Answer 1

To find the limit of the function √x / ln(x) as x approaches infinity, we can use l'Hospital's Rule. Applying l'Hospital's Rule twice, we find that the limit is 0/0.

Step-by-step explanation:

To find the limit of the function √x / ln(x) as x approaches infinity, we can use l'Hospital's Rule. Before applying l'Hospital's Rule, let's simplify the function by multiplying both the numerator and denominator by 1/√x. This gives us the equivalent function (1/√x) * (√x / ln(x)). Now, we can apply l'Hospital's Rule.

Taking the derivative of the numerator and denominator separately, we get (1/2) * (1/sqrt(x)) and (1/x * ln(x)). Taking the limit of these derivatives as x approaches infinity, we find that the limit is 0/∞.

Since we have an indeterminate form of 0/∞, we can apply l'Hospital's Rule again. After taking the derivatives, we get (1/2) * (1/2 * -1/x^2) and ((1/x) * (1/x) + ln(x)/x * -1/x^2). Taking the limit of these derivatives as x approaches infinity, we get 0/0.