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When you've got a limit that looks like 0/0 or ∞/∞, L'Hôpital's rule can often find its value -- and make it clear that not all infinities are equal!

Evaluate \(\displaystyle \lim_{x \to 0} \frac{e^{14x}- 1 - 14x}{x^2}\).

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Evaluate \[\lim_{x \to 0^+} (1+\sin 6x)^{\cot x}.\]

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Evaluate \[\lim_{x \to 0} \frac{e^{2x}-1-2x}{x^2}.\]

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Evaluate \[\lim_{x \to 0} \frac{6\tan x-6x}{x^3}.\]

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Evaluate \[\lim_{x \to 1} \frac{x^{9}-9x+8}{(x-1)^2}.\]

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