Find the limit as a approaches t. ?

Question

(a^5-t^5)/(a^2-t^2)

Answer 1

The simplest way to seek the solution is to use L'Hospital's Rule (LHR) which allows you to find the limit for this case by evaluating the ratio of the derivative of the numerator (N) and the derivative of the denominator (D), both taken with respect to the variable (a), while treating t as a constant. To illustrate the LHR method:

(1) Given: (a^5-t^5)/(a^2-t^2)

(2) N(a) = a^5 - t^5 and D(a) = a^2 - t^2

(3) Then dN/da = 5*a^4 and dD/da = 2*a

(Remember: parameter t is a constant, a non-variable)

(4) The ratio is (5/2)*a^3

(5) Now, applying the limit that a approaches t, we find

that the limit is (5/2)*t^3

Note: If you are skeptical about the validity of this, try some numbers to see that it works. For example, try with t = 10 and pick an a-value close to 10, something like 10.0001 to verify that the result is very close to the above-prescribed limit of 2.5*10^3 = 2500. And try others on your computer/calculator to gain more confidence, as you like.