Using the definition of limit at infinity or infinite limits, prove that
a) limπ₯ β 3 1/ (π₯ β 3) 2 = β
Let C>0C>0C>0 be fixed. Solve
β£1(xβ3)2β£>Cββ£xβ3β£<1C\left| \frac{1}{\left( x-3 \right) ^2} \right|>C\Leftrightarrow \left| x-3 \right|<\frac{1}{\sqrt{C}}β£β£β(xβ3)21ββ£β£β>Cββ£xβ3β£<Cβ1β
Thus for every C>0 there exists Ξ΄=1C\delta =\frac{1}{\sqrt{C}}Ξ΄=Cβ1β such that β£xβ3β£<Ξ΄ββ£1(xβ3)2β£>C\left| x-3 \right|<\delta \Rightarrow \left| \frac{1}{\left( x-3 \right) ^2} \right|>Cβ£xβ3β£<Ξ΄ββ£β£β(xβ3)21ββ£β£β>C
By the definition
limβ‘xβ31(xβ3)2=β\underset{x\rightarrow 3}{\lim}\frac{1}{\left( x-3 \right) ^2}=\inftyxβ3limβ(xβ3)21β=β
Need a fast expert's response?
and get a quick answer at the best price
for any assignment or question with DETAILED EXPLANATIONS!
Comments