if lim f(x) = a^3
x->a
and if lim g(x) = a^2
x->a
calculate the following limit:
lim f(x)*(x-a)/(x^3-a^3)*g(x) =
x->a
To calculate the limit:
1. Start by simplifying the expression. Notice that (x^3 - a^3) can be factored using the difference of cubes formula:
x^3 - a^3 = (x - a)(x^2 + ax + a^2)
2. Rewrite the expression using the factored form:
lim [f(x)*(x-a)/(x^3 - a^3)*g(x)] =
lim [f(x)*(x-a)/[(x - a)(x^2 + ax + a^2)]*g(x)]
3. Simplify the expression by canceling out the common factors (x - a):
lim [f(x)/[x^2 + ax + a^2]*g(x)]
4. Since we are taking the limit as x approaches a, we can substitute a for x in the expression:
f(a)/[a^2 + a(a) + a^2]*g(a)
5. Substitute the given limits for f(x) and g(x):
(a^3)/[a^2 + a(a) + a^2]*(a^2)
6. Simplify the expression:
(a^3)/(3a^2)*(a^2)
= a^3 / 3
Therefore, the limit of f(x)*(x-a)/(x^3 - a^3)*g(x) as x approaches a is a^3/3.