The statement means that for each there corresponds a such that if then If then Use a

Chapter 1, Problem 70

(choose chapter or problem)

Graphical Analysis The statement

\(\lim_{x\rightarrow3}\ \frac{x^2-3x}{x-3}\)

means that for each \(\varepsilon>0\) there corresponds a \(\delta>0\) such that if \(0<|x-3|<\delta\), then

\(\left|\frac{x^{2}-3 x}{x-3}-3\right|<\varepsilon\)

If \(\varepsilon=0.001\), then

\(\left|\frac{x^{2}-3 x}{x-3}-3\right|<0.001\)

Use a graphing utility to graph each side of this inequality. Use the zoom feature to find an interval \((3-\delta,\ 3+\delta)\) such that the graph of the left side is below the graph of the right side of the inequality.

Text Transcription:

lim_x rightarrow 3 x^2 - 3x / x-3

varepsilon > 0

delta > 0

0 < |x - 3| < delta

|x^2 - 3x / x-3 - 3| < varepsilon

varepsilon = 0.001

|x^2 - 3x / x-3 - 3| < 0.001

(3-delta, 3+delta)