Evaluate the following limit:

e) lim f(x)-f(a)/ x - a
x-> a

if f(x) = x^2 + 5

To evaluate the given limit, we need to substitute the given function f(x) = x^2 + 5 into the expression and simplify it.

Step 1: Substitute f(x) = x^2 + 5 into the expression:
lim (x -> a) [(f(x) - f(a)) / (x - a)]

Step 2: Substitute f(x) with its given expression:
lim (x -> a) [(x^2 + 5 - f(a)) / (x - a)]

Step 3: Substitute f(a) with its given expression:
lim (x -> a) [(x^2 + 5 - a^2 - 5) / (x - a)]

Step 4: Simplify the expression:
lim (x -> a) [(x^2 - a^2) / (x - a)]

Step 5: Factor the numerator:
lim (x -> a) [(x + a)(x - a) / (x - a)]

Step 6: Cancel out the common factor (x - a) in the numerator and denominator:
lim (x -> a) (x + a)

Step 7: Substitute x with a:
lim (x -> a) (a + a)

Step 8: Simplify the expression:
lim (x -> a) (2a)

Finally, we have evaluated the limit, which is 2a.