Question

find \\( \lim _{h \
ightarrow 0} \frac{f(x+h)-f(x)}{h} \\) for the given function and value of \\( x \\).
\\( f(x)=8 x-3, x=1 \\)
the \\( \lim _{h \
ightarrow 0} \frac{f(x+h)-f(x)}{h} \\) for \\( f(x)=8 x-3, x=1 \\) is
(type an integer or a simplified fraction.)

Explanation:

Step1: Find \(f(x + h)\)

Given \(f(x)=8x - 3\), then \(f(x + h)=8(x + h)-3=8x+8h - 3\).

Step2: Substitute into the difference - quotient formula

\(\frac{f(x + h)-f(x)}{h}=\frac{(8x + 8h-3)-(8x - 3)}{h}\).
Simplify the numerator: \((8x + 8h-3)-(8x - 3)=8x + 8h-3 - 8x + 3=8h\).
So, \(\frac{f(x + h)-f(x)}{h}=\frac{8h}{h}\), and for \(h
eq0\), \(\frac{8h}{h}=8\).

Step3: Find the limit as \(h

ightarrow0\)
Since \(\frac{f(x + h)-f(x)}{h}=8\) for \(h
eq0\), \(\lim_{h
ightarrow0}\frac{f(x + h)-f(x)}{h}=\lim_{h
ightarrow0}8\).
The limit of a constant function \(y = C\) as \(h
ightarrow a\) is \(C\), so \(\lim_{h
ightarrow0}8 = 8\).

Answer:

\(8\)