Q. 18 - Calculus

Question

In Example 2 we proved that $\underset{x\to 2}{\lim }\left(x^2-4x+5\right)=1$. Use the proof to find values of $\delta$ corresponding to

Part (a):$\in =1$

Part (b):$\in =0.1$

Part (c):$\in =0.01$

Illustrate that your choices of $\delta$ work by examining a graph of $f\left(x\right)=x^2-4x+5$and sketching appropriate $\in$ and $\delta$ intervals.

Step-by-Step Solution

Answer

Part (a): $\delta =1$

Part (b): $\delta =0.316$

Part (c): $\delta =0.1$

Part (a) Step 1. Given information.

Consider the given question,

$$\begin{gathered} \underset{x\to 2}{\lim }\left(x^2-4x+5\right)=1 \\ \in >0 \\ \delta =\sqrt{\in } \end{gathered}$$

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

Part (a) Step 2. Substitute &#8712; = 1 in the given equation.

It can be written,

$$\begin{gathered} =\left|\left(x^2-4x+5\right)-1\right| \\ =\left|x^2-4x+4\right| \\ ={\left|x-2\right|}^2 \\ ={\delta }^2 \\ ={\left(\sqrt{\in }\right)}^2 \\ =\in \end{gathered}$$

When $\in =1$, then,

$$\begin{gathered} \delta =\sqrt{1} \\ =1 \end{gathered}$$

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 $\left(1,2\right)\cup \left(2,3\right)$ has an $f\left(x\right)$ value in $\left(0,2\right)$.

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

It can be written,

$$\begin{gathered} =\left|\left(x^2-4x+5\right)-1\right| \\ =\left|x^2-4x+4\right| \\ ={\left|x-2\right|}^2 \\ ={\delta }^2 \\ ={\left(\sqrt{\in }\right)}^2 \\ =\in \end{gathered}$$

When $\in =0.1$, then,

$$\begin{gathered} \delta =\sqrt{0.1} \\ \approx 0.316 \end{gathered}$$

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 $(1.684,2)\cup (2,2.01)$ has an $f\left(x\right)$ value in $(0.9,1.1)$.

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

It can be written,

$$\begin{gathered} =\left|\left(x^2-4x+5\right)-1\right| \\ =\left|x^2-4x+4\right| \\ ={\left|x-2\right|}^2 \\ ={\delta }^2 \\ ={\left(\sqrt{\in }\right)}^2 \\ =\in \end{gathered}$$

When $\in =0.01$, then,

$$\begin{gathered} \delta =\sqrt{0.01} \\ =0.1 \end{gathered}$$

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 $(1.9,2)\cup (2,2.1)$ has an $f\left(x\right)$ value in $(0.99,1.01)$.