Evaluate the following limit:

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

if f(x) = x^2 + 5

Are you certain that is the problem? What you have is 0/0 when evaluated

Check the problem please, make certain you have it as assigned.

yes I know, it is supposed to give 0/0 and they ask us to manipulate the limit algebraically in order to be able to factorize

and it's the factorization that I can't seem to come up with, thanks for your quick answer too

= lim ( x^2 + 5 - a^2 - 5)/(x-a) , as x--> a

= lime (x+a)(x-a)/(x-a) , as x--> a
= lim 1/(x+a) , as x--> a
= 1/(2a)

To evaluate the given limit, we can start by plugging in the function f(x) = x^2 + 5 into the limit expression:

lim (x->a) [f(x) - f(a)] / [x - a]

Substituting f(x) = x^2 + 5:

lim (x->a) [(x^2 + 5) - (a^2 + 5)] / [x - a]

Simplifying the numerator:

lim (x->a) (x^2 + 5 - a^2 - 5) / (x - a)

Combining like terms:

lim (x->a) (x^2 - a^2) / (x - a)

Now, we notice that the numerator can be factored as a difference of squares:

lim (x->a) [(x - a)(x + a)] / (x - a)

We can then cancel out the common factor of (x - a):

lim (x->a) (x + a)

Finally, we substitute x = a into the expression:

lim (x->a) (a + a) = lim (x->a) 2a = 2a

Therefore, the limit is 2a.