Calculus

L'Hôpital's Rule

L'Hopital's Rule: Level 2 Challenges

         

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

limx0cotxcscx=? \displaystyle \lim_{x \to 0} \frac {\cot x }{\csc x } = \, ?

Details and Assumptions:

  • ddx(cotx)=csc2x \frac{d}{dx}( \cot x ) = -\csc^2 x
  • ddx(cscx)=cscxcotx \frac{d}{dx}( \csc x ) = -\csc x \cot x

limx1xn+xn1++xnx1= ?\large \lim_{x \to 1}\frac {x^{n} + x^{n-1} + \ldots + x - n}{x - 1} = \ ?

If the limit

limx0(sin2xx3+a+bx2)=0\lim _{ x\rightarrow 0 }{ \left( \frac { \sin { 2x } }{ { x }^{ 3 } } +a+\frac { b }{ { { x }^{ 2 } } } \right) } =0

is true for constants aa and bb, then what is the value of 3a+b?3a+b?

limx0(1x40x2eu2dux6)=? \lim_{x \to 0} \left(\frac{1}{x^4} - \frac{ {\displaystyle\int_{0}^{x^2}} e^{-u^2} du}{x^6}\right) = \, ?

What is the value of

limx0tanxxsinxx? \lim _{ x \rightarrow 0 } \frac{ \tan x - x } { \sin x - x }?

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