L’Hospital’s Rule and Indeterminate Forms — Question 6

Question 6 - Solution

As $x \to \infty$, we observe: $$x \to \infty, \quad \ln x \to \infty$$

So the limit is of the form: $$\infty - \infty \quad \text{(indeterminate form)}$$

To apply L’Hospital’s Rule, we rewrite the expression to form a quotient. Factor: $$x - \ln x = x \left( 1 - \frac{\ln x}{x} \right)$$

Now evaluate: $$\lim_{x \to \infty} \frac{\ln x}{x} = 0 \quad \text{(since exponential grows faster)}$$

So: $$\lim_{x \to \infty} \left( x - \ln x \right) = \infty \cdot (1 - 0) = \infty$$

Final Answer: $\boxed{\infty}$