Quiz 4 (Solution)

Date: MATH 1210-4 - Spring 2004

1. $\lim_{x\rightarrow 5} \frac{2x^2-11x+5}{x-5} = 9$. Give an $\epsilon$-$\delta$ proof of this fact.

   Given $\epsilon>0$ we have to find $\delta>0$ so that if $0< \vert x-5\vert <\delta$, then $\vert\frac{2x^2-11x+5}{x-5} -9\vert<\epsilon$.

   We have

   $$\begin{split}\vert\frac{2x^2-11x+5}{x-5} -9\vert &= \vert\fra... ...2 \vert\frac{(x-5)^2}{x-5}\vert = 2 \vert x-5\vert. \end{split}$$

   
   
   Now, $\vert x-5\vert<\delta$, so that $2\vert x-5\vert<\epsilon$ if we take $\delta =\epsilon/2$.

   All together, for $\delta =\epsilon/2$, if $0<\vert x-5\vert<\delta =\epsilon/2$ we have $\vert\frac{2x^2-11x+5}{x-5} -9\vert = 2\vert x-5\vert <2 \epsilon/2 = \epsilon$.

2. Knowing that $\lim_{x\rightarrow 7} f(x) = 1$ and $\lim_{x\rightarrow 7} g(x) = 9$, use properties of the limit operation to find

   $$\lim_{x\rightarrow 7} \frac{2 (g(x)-4f(x))^2}{\sqrt{g(x)}}.$$

   
   

   Since the expression is a quotient, we compute the limit of the quotient as the quotient of the limits, provided that the limit in the denominator is $\neq 0$ (this will become clear in the computation).

   $$\begin{split}\lim_{x\rightarrow 7} \frac{2(g(x)-4f(x))^2}{\sq... ...-4\cdot 1)^2}{3} = \frac{2(5)^2}{3} = \frac{50}{3}. \end{split}$$