Learn Decimals with Plainmath's In-Depth Explanations and Real-World Examples

(a) Write each answer as an equality, and any decimals up to 7 decimal places. Clearly state the value of and ${\displaystyle \delta \left(\text{or}M\text{or}N\text{and}\delta \right)}$ in each case.
(i) How close to 4 do we need to take x so that ${\displaystyle (\frac{x}{2}-2)<0.001}$ ?
(ii) How close to 4 do we need to take x so that ${\displaystyle (\frac{x}{2}-2)>0.0001}$ ?
(iii) How close to 0 do we need to take x so that ${\displaystyle (2x+9)<9.0001}$ ?
(iv) How close to 0 do we need to take x so that ${\displaystyle (2x+9)>8.999}$?
(v) How close to 0 do we need to take x so that ${\displaystyle (x^2+6x+9)<9.001}$ ?
(vi) How close to 0 do we need to take x so that ${\displaystyle (x^2+6x+9)<9.0001}$ ?
(vii) How close to 0 do we need to take x so that ${\displaystyle (x^2+6x+9)>8.9999}$ ?
(viii) How close to -7 do we need to take x so that ${\displaystyle \frac{1}{{(x+7)}^4}>10000}$ ?
(ix) How close to −7 do we need to take x so that ${\displaystyle \frac{1}{{(x+7)}^4}>100000}$ ?
(x) How close to 0 do we need to take x so that ${\displaystyle \ln x<-10000}$?
(b) Use the definition of a limit to show that
(i) ${\displaystyle limx\to 4}$
${\displaystyle (\frac{x}{2}-2)=0}$
(ii) ${\displaystyle limx\to 0}$
${\displaystyle (2x+9)=9}$
(iii) ${\displaystyle limx\to 0}$
${\displaystyle (x^2+6x+9)=9}$
(iv) ${\displaystyle limx\to -7}$
${\displaystyle \frac{1}{{(x+7)}^4}=\mathrm{\infty }}$
(v) ${\displaystyle limx\to 0+}$
${\displaystyle \ln x=-\mathrm{\infty }}$

A simplified form of ${\displaystyle \left(\surd 3^{2}x\right)+2+3^{-2}x}$

Find the sum of the series to the nearest four decimal places.
${\displaystyle \sum _{n=1}^{\mathrm{\infty }}\frac{{(-1)}^n}{\left(2n\right)!}}$

Solving Inequalities Graphically: Use a graphing device to solve the inequality. Express your answer using interval notation, with the endpoints of the interval rounded to two decimals.
${\displaystyle 2x^3-3x+1<0}$