Answer
$m=3$
Work Step by Step
$f(x)=3x+4,$ at $(1,7)$
The slope of the tangent line at the point $P(a,f(a))$ is given by $m=\lim_{h\to0}\dfrac{f(a+h)-f(a)}{h}$
In this case, $a=1$.
Find $f(a+h)$ by substituting $x$ by $1+h$ in $f(x)$ and simplifying:
$f(1+h)=3(1+h)+4=3+3h+4=7+3h$
Find $f(a)$ by substituting $x$ by $1$ in $f(x)$ and simplifying:
$f(1)=3(1)+4=3+4=7$
Substitute the known values into the formula that gives the slope of the tangent line and evaluate:
$m=\lim_{h\to0}\dfrac{f(1+h)-f(1)}{h}=\lim_{h\to0}\dfrac{(7+3h)-7}{h}=...$
$...=\lim_{h\to0}\dfrac{3h}{h}=\lim_{h\to0}3=3$
