find the limit as h approaches 0

Question

find lim_h→0 {[(27+h)^(1/3)] - 3} / h

Answer 1

Hi Daniel, My guess is that you perhaps have not learned how to find the derivative of a function. The intent of the problem likely is to evaluate the limit directly. If that is the case:

 

The idea is to find something we can multiply numerator and denominator by so that the "h" in the denominator will cancel out and when you substitute 0 for h you will no longer get 0/0.

 

Do you recall how to factor a difference of cubes? (a³ - b³) = (a - b)(a² +ab + b²).

 

The numerator of your limit problem is sort of like the (a + b) in the above, i.e. each term is the cube root of something. If you can format the trinomial to follow the pattern above that corresponds to ((27+h)^1/3 - 3)), that is what you want to multiply top and bottom of your difference quotient by.

 

That trinomial is found by following the pattern: square the first term, take the opposite of the product of the two terms, square the 2nd term).

 

So your multiplier looks like this: [(27+h)^2/3 +3(27+h)^1/3 + 9]. When you multiply the numerator by this expression the result will be a difference of cubes (but the cubes will not be apparent because of working with fractional exponents).

 

The new numerator should end up looking line this [(27 + h) - 27] = h.

 

The new denominator will look like this h [(27+h)^2/3+3(27+h)^1/3 + 9]. The h in the numerator cancels out, and you can find the limit as h->0 by direct substitution. Try it and see what you get.

Answer 2

Take the derivative of f(x) = x^1/3       at x=27.