Limits Help

Question

 

Im pretty sure the first one is a composite limit and the next i just have to find the limit

 

lim(x→0) [cos?(x)]/[ln?(sin?(x) )]

 

 

lim(x→∞) [(e^x)/(1-e^x )] 

 

 

lim(x→-∞) [(x^2)/√(2x^4-3x)]

Answer 1

I don't understand what you mean by the first one, but here are solutions for the other two:

 

lim_x→∞[e^x/(1-e^x)] has the indeterminate form ∞/(-∞)

 

= lim_x→∞[1/((1/e^x) - 1)] = 1/(0-1) = -1

 

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lim_x→-∞[x²/√(2x⁴-3x)] has the indeterminate form ∞/∞

 

  √(2x⁴-3x) can be written as √(x⁴)√(2-(3x/x⁴)) = x² √(2 - (3/x³))

 

So, the given limit is equivalent to lim_x→∞[1/√(2 - (3/x³))] = 1/(√2-0)

 

                                                                                   = 1/√2