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L'Hôpital's Rule

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!

Level 2

         

Please help me! I kept using L'Hôpital's rule millions of times and I can't evaluate the limit below!

\[ \displaystyle \lim_{x \to 0} \frac {\cot x }{\csc x } = \ ? \]

Details and assumptions:

  • \( \frac{d}{dx}( \cot x ) = -\csc^2 x \)

  • \( \frac{d}{dx}( \csc x ) = -\csc x \cot x \)

\[\large \lim_{x \to 1}\frac {x^{n} + x^{n-1} + \ldots + x - n}{x - 1} = \ ? \]

If the limit

\[\lim _{ x\rightarrow 0 }{ \left( \frac { \sin { 2x } }{ { x }^{ 3 } } +a+\frac { b }{ { { x }^{ 2 } } } \right) } =0\]

is true for constants \(a\) and \(b\), then what is the value of \(3a+b?\)

\[ \large{ \lim_{x \to 0}}\large \left(\frac{1}{x^4} - \frac{\displaystyle \int_{0}^{x^2} e^{-u^2} du}{x^6}\right) = \ ? \]

What is the value of

\[ \lim _{ x \rightarrow 0 } \frac{ \tan x - x } { \sin x - x }? \]

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