A function $\mathrm { f } ( \mathrm { x } )$, a point $\mathrm { c }$, the limit of $\mathrm { f } ( \mathrm { x } )$ as $x$ approaches $\mathrm { c }$, and a positive number $\varepsilon$ is given. Find a number $\delta 0$ such that for all $x , 0

Question

A function $\mathrm { f } ( \mathrm { x } )$, a point $\mathrm { c }$, the limit of $\mathrm { f } ( \mathrm { x } )$ as $x$ approaches $\mathrm { c }$, and a positive number $\varepsilon$ is given. Find a number $\delta > 0$ such that for all $x , 0 < | \mathrm { x } - \mathrm { c } | < \delta \Rightarrow | \mathrm { f } ( \mathrm { x } ) - \mathrm { L } | < \varepsilon$.
-$$f ( x ) = \frac { 14 } { x } , c = 7 , \varepsilon = 0.4$$
A) $L = 2 ; \delta = 3.5$
B) $L = 2 ; \delta = 1.75$
C) $\mathrm { L } = 2 ; \delta = 1.17$
D) $L = 2 ; \delta = 17.5$

Quizplus · Accepted Answer

We use the definition of the limit to find the appropriate delta. First, we find the limit of $f(x)$ as $x$ approaches $c=7$: $$\lim_{x\to7}\frac{14}{x}=2.$$ Next, we want to find a delta such that for all $x,$ if $0<|x-7|<\delta,$ then $$\left|\frac{14}{x}-2\right|<0.4.$$ Taking the absolute value of the difference gives us $$\left|\frac{14}{x}-2\right|=\left|\frac{14-2x}{x}\right|.$$ We want this to be less than 0.4, so we have the inequality $$\left|\frac{14-2x}{x}\right|<0.4.$$ Multiplying both sides by $|x|$ (and remembering to flip the inequality since $x$ can be negative) gives us $$|14-2x|<0.4|x|.$$ The left-hand side is a linear function, so it achieves its maximum when $x$ is as far away from 7 as possible. We can assume without loss of generality that $x>7$ (since if $x<7,$ we can use the same logic with $-x$ instead of $x$). Then we have $$14-2x=14-2(x-7)=28-2x,$$ so the inequality becomes $$28-2x<0.4x.$$ Simplifying and solving for $x,$ we get $x<14. $ Similarly, if we assume that $x<7,$ we get $x>0.$ So we need to choose a delta that satisfies both $7-\delta>0$ and $7+\delta<14.$ The largest such delta is $\delta=1.75,$ which is given in choice (B).

A Function f(x)\mathrm { f } ( \mathrm { x } )f(x)

A Function $\mathrm { f } ( \mathrm { x } )$