Q. 6 - Calculus

Question

Suppose $$\begin{gathered} f\left(x\right)=2^x-4, \\ g\left(x\right)=x-2 \end{gathered}$$. Find the equations of the tangent lines to these functions at $x=2$. Then argue graphically that it would be reasonable to think that the limit of the quotient $\frac{f\left(x\right)}{g\left(x\right)}$ as $x\to 2$ might be equal to the limit of the quotient of these tangent lines as $x\to 2$.

Step-by-Step Solution

Answer

$y=2.8x-5.6$ is the line tangent to $f\left(x\right)$ when x is $2$.

$y=x-2$ is the line tangent to $g\left(x\right)$ when x is $2$.

Very close to $x=2$ the graphs of $f,g$ are very close to the graphs of the given tangent lines.

Step 1. Given information.

Consider the given question,

$$\begin{gathered} f\left(x\right)=2^x-4, \\ g\left(x\right)=x-2 \end{gathered}$$

Step 2. Consider the function f x .

Consider the function $f\left(x\right)=2^x-4$.

The point of tangency is given below,

$$\begin{gathered} f\left(2\right)=2^2-4 \\ =0 \\ f\left(2\right)\to \left(2,0\right) \end{gathered}$$

Slope of tangent, $f\text{'}\left(x\right)=\ln 2·2^x$.

Then the slope of tangent at $x=2$,

$$\begin{gathered} f\text{'}\left(2\right)=\ln 2·2^2 \\ =2.8 \end{gathered}$$

So $y=mx+c$.

Substitute the values in the above equation,

$$\begin{gathered} 0=2.8\times 2+c \\ c=-5.6 \\ y=2.8x-5.6 \end{gathered}$$

Step 3. Consider the function g x .

Consider the function $g\left(x\right)=x-2$.

The point of tangency is given below,

$$\begin{gathered} f\left(2\right)=2-2 \\ =0 \\ f\left(2\right)\to \left(2,0\right) \end{gathered}$$

Slope of tangent, $g\text{'}\left(x\right)=1$.

Then the slope of tangent at $x=2$,

$g\text{'}\left(2\right)=1$

So $y=mx+c$.

Substitute the values in the above equation,

$$\begin{gathered} 0=1\times 2+c \\ c=-2 \\ y=x-2 \end{gathered}$$

Step 4. Plot the graphs.

The two tangent lines are $$\begin{gathered} y=2.8x-5.6 \\ y=x-2 \end{gathered}$$.

On plotting the graph,

From the graph, we can say the limit of the quotient of tangent lines as $x\to 2$is equal to the limit of the quotient $\frac{f\left(x\right)}{g\left(x\right)}$ as $x\to 2$.