Provide an appropriate response.
-Given $\varepsilon > 0$ , find an interval $I = ( 5 - \delta , 5 ) , \delta > 0$ , such that if $x$ lies in $I$ , then $\sqrt { 5 - x } < \varepsilon$ . What limit is being verified and what is its value?

Question

Provide an appropriate response.
-Given $\varepsilon > 0$, find an interval $I = ( 5 - \delta , 5 ) , \delta > 0$, such that if $x$ lies in $I$, then $\sqrt { 5 - x } < \varepsilon$. What limit is being verified and what is its value?
A) $\lim _ { x \rightarrow 5 ^ { - } } \sqrt { x } = 5$
B) $\lim _ { x - 5 ^ { + } } \sqrt { 5 - x } = 0$
C) $\lim _ { x \rightarrow 0 ^ { - } } \sqrt { 5 - x } = 0$
D) $\lim _ { x \rightarrow 5 ^ { - } } \sqrt { 5 - x } = 0$

Quizplus · Accepted Answer

The given interval is $I = ( 5 - \delta , 5 )$ and we need to show that $\sqrt { 5 - x } < \varepsilon$ if $x$ lies in $I$. This means that we want to show that $\lim _ { x \rightarrow 5 ^ { - } } \sqrt { 5 - x } = 0$, since the definition of a limit says that for any $\varepsilon > 0$, we can find a $\delta > 0$ such that if $0 < 5 - x < \delta$, then $\sqrt { 5 - x } < \varepsilon$. Therefore, the correct choice is D.

Provide an Appropriate Response ε>0\varepsilon > 0ε>0 , Find an Interval I=(5−δ,5),δ>0I = ( 5 - \delta , 5 ) , \delta > 0I=(5−δ,5),δ>0

Provide an Appropriate Response $\varepsilon > 0$ , Find an Interval $I = ( 5 - \delta , 5 ) , \delta > 0$