Math Snap

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 $2.5$.
$$\delta = \min(d_A, d_B)$$

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