Question

Find f(a), f(a + h), and the difference quotient

Find f(a), f(a + h), and the difference quotient

f(a + h) − f(a)
h
Where
h ≠ 0. f(x) = 5x + 6

f(a) =
f(a + h) =
{f(a + h) − f(a)}/h=

what are the steps to solve the difference quotient
{f(a + h) − f(a)}/h, I have tried but no luck.

Answer 1

Well, f(a) you just plug straight into the equation! Thus, f(a) = 5a + 6.
The same goes for f(a+h)! Thus, f(a+h) = 5a + 5h + 6
Your derivative (expressed as the difference quotient) is
\[\lim_{h \rightarrow 0}\frac{ f(a+h) - f(a) }{ h } = \frac{ 5a+5h+6 - 5a - 6 }{ h }\]
\[=\lim_{h \rightarrow 0}\frac{ 5h }{ h } = 5.\]

Answer 2

Extra: via the definition of the limit, the function does need to exist at h = 0--just a surrounding neighborhood. That is why I am allowed to divide h out without worrying about the potential that h could be zero.