What is the limit as x approaches 0 of ( f(x^2+3x) divided by sin(x) ) ? Given that…(see description)

Question

GIVEN:  f(x) is a differentiable function with the properties that the limit as x approaches 0 of f(x^2+3x) is 0 and the limit as x approaches 0 of f'(x^2+3x) is 4

Answer 1

Hi Matt,

 

This problem is a complicated way of writing a limit involving indeterminate forms, L'Hopital's rule, and the chain rule for derivatives.

 

lim_x→0 f(x²+3x) / sin(x)

 

If you take the limit as it is, you get 0 / 0, which is indeterminate. You can often use L'Hopital's rule to find the limit, by taking the derivatives of the top and bottom separately. Don't forget the chain rule on the top.

 

lim_x→0 f(x²+3x) / sin(x)

= lim_x→0 f'(x²+3x)*(2x+3) / cos(x)

 

The limit of the derivative is given as 4, so this all equals

= 4*(2*0+3) / 1 = 12