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Chapter 3: Q. 6 (page 310)

Suppose $f (x) = 2^{x} - 4, g (x) = x - 2$ . 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 $\frac{f (x)}{g (x)}$ as $x 鈫� 2$ might be equal to the limit of the quotient of these tangent lines as $x 鈫� 2$ .

Short Answer

Expert verified

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

$y = x - 2$ is the line tangent to $g (x)$ 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 by step solution

01

Step 1. Given information.

Consider the given question,

$f (x) = 2^{x} - 4, g (x) = x - 2$

02

Step 2. Consider the function fx.

Consider the function $f (x) = 2^{x} - 4$ .

The point of tangency is given below,

$f (2) = 2^{2} - 4 = 0 f (2) 鈫� (2, 0)$

Slope of tangent, $f' (x) = \ln 2 路 2^{x}$ .

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

$f' (2) = \ln 2 路 2^{2} = 2.8$

So $y = m x + c$ .

Substitute the values in the above equation,

$0 = 2.8 脳 2 + c c = - 5.6 y = 2.8 x - 5.6$

03

Step 3. Consider the function gx.

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

The point of tangency is given below,

$f (2) = 2 - 2 = 0 f (2) 鈫� (2, 0)$

Slope of tangent, $g' (x) = 1$ .

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

$g' (2) = 1$

So $y = m x + c$ .

Substitute the values in the above equation,

$0 = 1 脳 2 + c c = - 2 y = x - 2$

04

Step 4. Plot the graphs.

The two tangent lines are $y = 2.8 x - 5.6 y = x - 2$ .

On plotting the graph,

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

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