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

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

limx1xa1xb1\lim_{x\rightarrow 1} \frac{x^a - 1}{x^b - 1}

Compute the following limit

limx0sin(5x)x\lim_{x\rightarrow 0} \frac{\sin{(5x)}}{x}

Use l'Hospital's Rule to find the limit

limx0x26x+2x+1\lim_{x\rightarrow 0} \frac{x^2 - 6x + 2}{x + 1}

Evaluate the following limit

limxln(1+e3x)2x+5\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.

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

Evalute limx0x+cos(2x)exx\lim_{x\rightarrow 0} \frac{x + \cos{(2x)} - e^x}{x}

Evaluate the following limit

limθ0tan(θ)θθsin(θ)\lim_{\theta\rightarrow 0} \frac{\tan{(\theta)} - \theta}{\theta - \sin{(\theta)}}

Evaluate the following limit

limx0sin(3x)sin(4x)\lim_{x\rightarrow 0} \frac{\sin{(3x)}}{\sin{(4x)}}