Given f(x) = x^3, evaluate lim h-->o f(a+h)-f(a)/h
To evaluate the limit of the given expression, we can start by using the limit definition of the derivative. Here's how:
Step 1: Write down the expression we want to evaluate the limit of:
lim h-->0 [f(a+h) - f(a)] / h.
Step 2: Substitute the given function f(x) = x^3 into the expression:
lim h-->0 [(a+h)^3 - a^3] / h.
Step 3: Expand (a+h)^3 using the binomial expansion formula:
lim h-->0 [(a^3 + 3a^2h + 3ah^2 + h^3) - a^3] / h.
Step 4: Simplify the expression by canceling out the common terms:
lim h-->0 [3a^2h + 3ah^2 + h^3] / h.
Step 5: Factor out an h from the numerator:
lim h-->0 h(3a^2 + 3ah + h^2) / h.
Step 6: Cancel out h in the numerator and denominator:
lim h-->0 3a^2 + 3ah + h^2.
Step 7: Since we are taking the limit as h approaches 0, we can substitute h = 0 into the expression:
3a^2 + 3a * 0 + 0^2.
Step 8: Simplify the expression:
3a^2 + 0 + 0 = 3a^2.
Therefore, the limit of the expression lim h-->0 [f(a+h) - f(a)] / h is 3a^2.