QuestionFind $\delta$ so that if $0<|x-3|<\delta$ , then $|f(x)-2|<0.5$ using the graph of $f$ .

Studdy Solution

STEP 1

Assumptions1. We are given a graph of function $f$ . . We need to find a number $\delta$ such that if $0<|x-3|<\delta$ then $|f(x)-|<0.5$ .
3. The function $f$ is continuous at $x=3$ .

STEP 2

The problem is asking us to find a $\delta$ such that for all $x$ in the interval $(-\delta,+\delta)$ , the values of $f(x)$ are in the interval $(2-0.5,2+0.5)$ , or $(1.5,2.5)$ .

STEP 3

On the graph, locate the points where $f(x)$ intersects the lines $y=1.5$ and $y=2.5$ . Let's call these points $A$ and $B$ respectively.

STEP 4

Find the $x$ -coordinates of points $A$ and $B$ . Let's denote them as $x_A$ and $x_B$ respectively.

STEP 5

Calculate the distances from $x=3$ to $x_A$ and $x_B$ .
$d_A = |3 - x_A|$ $d_B = |3 - x_B|$

STEP 6

The value of $\delta$ we are looking for is the smaller of $d_A$ and $d_B$ . This is because we want the interval $(3-\delta,3+\delta)$ to be completely contained within the interval where $f(x)$ is between $1.5$ and $2.5$ .
$\delta = \min(d_A, d_B)$

STEP 7

Now we have found the value of $\delta$ that satisfies the given condition.

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