Evaluate the limit

Question

lim (h --> 0)   (√(1 + h) -1)/h

 

 

ANSWER: 1/2

#14, practice problems

Answer 1

My guess is that this problem was not intended to be solved by L'Hospital's Rule, i.e. it is likely the student has not reached a point in the course where L'Hospital's Rule has been introduced. But I could be wrong.

 

Another way to find the limit is to rewrite the function so that the form is not indeterminate.

 

That is done by multiplying both numerator and denominator of the original expression by the "conjugate" of the numerator. That would be √(1+h) + 1. Since conjugate multiplication results in a difference of squares the new numerator becomes (1+h) - 1 = h.

 

The new denominator becomes h(√(1+h) + 1).

 

So, now the problem looks like this:

 

lim_h->0 h/[ h(√(1+h) + 1)]. The "h" factor cancels out, and substitution gives the answer 1/(1+1) = 1/2.

Answer 2

We can use L'Hopsital Rule to find the limit.  Take the derivative of the numerator divided by the derivative of the denominator.

 

lim       (1/2)(1 + h)^-1/2 / 1

h-->0

 

 

lim          1 / (2√(1 + h))

h-->0

 

 

Plug in h=0.

 

 

lim         1 / 2

h-->0

 

 

As h approaches zero, the limit is 1 / 2.