Q. 19 - Calculus

Question

In Example 4 we proved that $\underset{x\to 2}{\lim }5x^4=80$. Use the proof to find values of $\delta$ corresponding to

Part (a):$\in =5$

Part (b):$\in =0.01$

Part (c):$\in =350$

Illustrate that your choices of $\delta$ work by examining a graph of $f\left(x\right)=5x^4$and sketching appropriate $\in$ and $\delta$ intervals.

Step-by-Step Solution

Answer

Part (a): $\delta =\frac{1}{65}$

Part (b): $\delta =\frac{0.01}{325}$

Part (c): $\delta =1$

Part (a) Step 1. Given information.

Consider the given question,

$$\begin{gathered} \underset{x\to 2}{\lim }5x^4=80 \\ \in >0 \\ \delta =\frac{\in }{325} \end{gathered}$$

For all x with $0<\left|x-2\right|<\delta$.

Part (a) Step 2. Substitute &#8712; = 5 in the given equation.

It can be written,

$$\begin{gathered} =\left|5\left(x^4-16\right)\right| \\ =\left|5\left(x-2\right)\left(x+2\right)\left(x^2+4\right)\right| \end{gathered}$$

Then,

$$\begin{gathered} <5\delta \left|\left(x+2\right)\left(x^2+4\right)\right| \\ =5\delta ·65 \\ =325\delta \\ =325\left(\frac{\in }{325}\right) \\ =\in \end{gathered}$$

When $\in =5$, then,

$$\begin{gathered} \delta =\frac{5}{325} \\ =\frac{1}{65} \end{gathered}$$

Part (a) Step 3. Sketching the function to illustrate the choice of &#948; .

Draw the graph of the given function and indicate on the graph that every value of x in $(1.985,2)\cup (2,2.015)$ has an $f\left(x\right)$value in $(75,85)$.

Part (b) Step 1. Substitute &#8712; = 0 . 01 in the given equation.

It can be written,

$$\begin{gathered} =\left|5\left(x^4-16\right)\right| \\ =\left|5\left(x-2\right)\left(x+2\right)\left(x^2+4\right)\right| \end{gathered}$$
Then,

$$\begin{gathered} <5\delta \left|\left(x+2\right)\left(x^2+4\right)\right| \\ =5\delta ·65 \\ =325\delta \\ =325\left(\frac{\in }{325}\right) \\ =\in \end{gathered}$$

When $\in =0.01$, then,

$\delta =\frac{0.01}{325}$ 

Part (b) Step 2. Sketching the function to illustrate the choice of &#948; .

Draw the graph of the given function and indicate on the graph that every value of x in $\left(-1.077,2\right)\cup \left(2,5.077\right)$ has an $f\left(x\right)$ value in $\left(79.99,80.01\right)$.

Part (c) Step 1. Substitute &#8712; = 350 in the given equation.

It can be written,

$$\begin{gathered} =\left|5\left(x^4-16\right)\right| \\ =\left|5\left(x-2\right)\left(x+2\right)\left(x^2+4\right)\right| \end{gathered}$$

Then,

$$\begin{gathered} <5\delta \left|\left(x+2\right)\left(x^2+4\right)\right| \\ =5\delta ·65 \\ =325\delta \\ =325\left(\frac{\in }{325}\right) \\ =\in \end{gathered}$$

When $\in =350$, then,

$$\begin{gathered} \delta =\frac{350}{325} \\ =1.077 \\ \approx 1 \end{gathered}$$

Part (c) Step 2. Sketching the function to illustrate the choice of &#948; .

Draw the graph of the given function and indicate on the graph that every value of x in $(1,2)\cup (2,3)$ has an $f\left(x\right)$ value in $(-270,430)$.