Work in a group of at most 4. Explain to those who do not understand. Ask questions if you do not understand.
- (20 points) Describe three situations where \(\displaystyle \lim_{x\rightarrow c} f(x)\) does not exist.
- (25 points) A machinist is required to
manufacture a circular metal disk with area \(1000\,
\mathrm{cm}^2\).
- What radius produces such a disc?
- If the machinist is allowed an error tolerance of \(\pm
3\, \mathrm{cm}^2\) in the area of the disk, how close to the
ideal radius you found above must the machinist control the
radius?
- In terms of the \(\epsilon\), \(\delta\) definition of
\(\displaystyle \lim_{x\rightarrow a} f(x)=L\), what is
\(x\)? What is \(a\)? What is \(L\)? What value of
\(\epsilon\) is given? What is the corresponding value of
\(\delta\)?
- (25 points)
- Define "\(\displaystyle \lim_{x\rightarrow c} f(x) =L\)".
- Using the definition, prove that \(\displaystyle \lim_{x\rightarrow 3} 5 x^2 = 45\).
- (30 points) Simplify and cancel so that you can plug in the given value
without dividing by 0. Then plug in the value.
- For \(x=2\), \( \frac{x^2+x-6}{x-2}=\)
- For \(x=4\), \( \frac{\sqrt{x}-2}{x-4}=\)
- For \(h=0\), \( \frac{(x+h)^2 -x^2}{h}=\)
- For \(h=0\), \( \frac{(x+h)^{-1} - x^{-1}}{h}=\)
- For \(h=0\) and \(f(x)=2x^2\), \(\frac{f(x+h) -f(x)}{h}=\)
Last modified: Sun Jan 2 20:07:14 UTC 2022