What is the limit of x as it approaches 0 for the function (sqrt(9 + h) - 3)/h

Question

I can't figure out how to get that pesky zero out of the denominator!

Answer 1

Hello again, Mia!

 

 

So your instinct is right - you can't get that "pesky" h out of the denominator. However, you can dilute it using the conjugate.

 

√(9+h)-3    *   √(9+h)+3

     h                √(9+h)+3

 

 

 

This works to clear up the numerator as well as dilute the denominator
     9+h-9     

h(√(9+h)+3

 

 

 

        h        
h(√(9+h)+3

 

 

 

        1        
√(9+h)+3

 

 

 

Now you can substitute that zero in without turning the expression into an undefined one (i.e. a zero in the denominator)

        1
√(9+0)+3

 

 

     1
√(9)+3

 

 

    1
  3+3

 

 

THUS,
lim h→0  = 1/6

 

Hope this helps!

 

Answer 2

lim_h→0[(√(9+h) - 3)/h]  Multiply the numerator and denominator by the                                    conjugate of the numerator.  

 

= lim_h→0[(√(9+h)-3)(√(9+h)+3)/(h(√(9+h)+3))]

 

= lim_h→0(9+h-9)/(h(√(9+h)+3))

 

= lim_h→0(1/(√(9+h)+3) = 1/(√9+3) = 1/6