Calculus 1: Indeterminate Forms and l'Hospital's Rule

Use l'Hospital's Rule to find the following limit

$\lim_{x\rightarrow 1} \frac{x^a - 1}{x^b - 1}$

Compute the following limit

$\lim_{x\rightarrow 0} \frac{\sin{(5x)}}{x}$

Use l'Hospital's Rule to find the limit

$\lim_{x\rightarrow 0} \frac{x^2 - 6x + 2}{x + 1}$

Evaluate the following limit

$\lim_{x\rightarrow \infty} \frac{\ln{(1 + e^{3x})}}{2x + 5}$

Explain why the following limit can not be found using l'Hospital's Rule then find the limit using a different method.

$\lim_{x\rightarrow \infty} \frac{x + \cos{(x)}}{x}$

Evalute $\lim_{x\rightarrow 0} \frac{x + \cos{(2x)} - e^x}{x}$

Evaluate the following limit

$\lim_{\theta\rightarrow 0} \frac{\tan{(\theta)} - \theta}{\theta - \sin{(\theta)}}$

Evaluate the following limit

$\lim_{x\rightarrow 0} \frac{\sin{(3x)}}{\sin{(4x)}}$