Q. 5 - Calculus

Question

Suppose $$\begin{gathered} f\left(x\right)=x^2-1, \\ g\left(x\right)=\ln x \end{gathered}$$. Find the equations of the tangent lines to these functions at $x=1$. 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 $x\to 1$ might be equal to the limit of the quotient of these tangent lines as $x\to 1$.

Step-by-Step Solution

Answer

$y=2\left(x-1\right)$ is the line tangent to $f\left(x\right)$ when x is $1$.

$y=x-1$ is the line tangent to $g\left(x\right)$ when x is $1$.

Very close to $x=1$ the graphs of $f,g$ are very close to the graphs of the given tangent lines.

Step 1. Given information.

Consider the given question,

The functions are $$\begin{gathered} f\left(x\right)=x^2-1, \\ g\left(x\right)=\ln x \end{gathered}$$.

Step 2. Consider the function f x .

Consider the function $f\left(x\right)=x^2-1$.

The point of tangency is given below,

$$\begin{gathered} f\left(1\right)=1^2-1 \\ =0 \\ f\left(1\right)\to \left(1,0\right) \end{gathered}$$

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

Then the slope of tangent at $x=1$,

$$\begin{gathered} f\text{'}\left(1\right)=2\left(1\right) \\ =2 \end{gathered}$$

So $y=mx+c$.

Substitute the values in the above equation,

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

Step 3. Consider the g x .

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

The point of tangency is given below,

$$\begin{gathered} g\left(1\right)=\ln \left(1\right) \\ =0 \\ g\left(1\right)\to \left(1,0\right) \end{gathered}$$

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

Then the slope of tangent at $x=1$,

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

So $y=mx+c$.

Substitute the values in the above equation,

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

Step 4. Plot the graphs.

The two tangent lines are $$\begin{gathered} y=2x-2 \\ y=x-1 \end{gathered}$$.

On plotting the graph, 

From the graph, we can say the limit of the quotient of tangent lines as $x\to 1$is equal to the limit of the quotient $\frac{f\left(x\right)}{g\left(x\right)}$ as $x\to 1$