Question

How do you use the limit definition to find the slope of the tangent line to the graph $f\left( x \right)=9x-2$ at $\left( 3,25 \right)$?

Answer 1

We start solving the problem by assuming the variable for the required slope of the tangent. We then make use of the fact that the slope of the tangent for any point on the given function $f\left( x \right)$ is defined as $\displaystyle \lim_{h \to 0}\dfrac{f\left( x+h \right)-f\left( x \right)}{h}$. We then make the necessary calculations to get the required value of slope of the given tangent.

Complete step by step answer:
According to the problem, we are asked to find the slope of the tangent line to the graph $f\left( x \right)=9x-2$ at $\left( 3,25 \right)$ using the limit definition.
Let us assume the slope of the tangent be m.
We know that the slope of the tangent for any point on the given function $f\left( x \right)$ is defined as $\displaystyle \lim_{h \to 0}\dfrac{f\left( x+h \right)-f\left( x \right)}{h}$.
So, we have $m=\displaystyle \lim_{h \to 0}\dfrac{f\left( x+h \right)-f\left( x \right)}{h}$.
$\Rightarrow m=\displaystyle \lim_{h \to 0}\dfrac{9\left( x+h \right)-2-\left( 9x-2 \right)}{h}$.
$\Rightarrow m=\displaystyle \lim_{h \to 0}\dfrac{9x+9h-2-9x+2}{h}$.
$\Rightarrow m=\displaystyle \lim_{h \to 0}\dfrac{9h}{h}$.
$\Rightarrow m=\displaystyle \lim_{h \to 0}9$.
$\Rightarrow m=9$.
We can see that the slope of the tangent at any point of the given graph $f\left( x \right)=9x-2$ is constant which is equal to 9.

$\therefore$ The slope of the tangent line to the graph $f\left( x \right)=9x-2$ at $\left( 3,25 \right)$ is 9.

Note: Here we have got constant slope for the tangent (i.e., independent of the variable x) otherwise, the values will be different at different points on the function. We can also find the equation of the tangent by making use of the fact that tangent is a straight line passing through the given point. Similarly, we can expect problems to find the equation of the tangent for $f\left( x \right)={{x}^{2}}+4x+2$ at $\left( 1,7 \right)$ using differentiation.