Question #313526

Using the definition of limit at infinity or infinite limits, prove that

a) limπ‘₯ β†’ 3 1/ (π‘₯ βˆ’ 3) 2 = ∞ 


1
Expert's answer
2022-03-19T02:39:39-0400

Let C>0C>0 be fixed. Solve

∣1(xβˆ’3)2∣>Cβ‡”βˆ£xβˆ’3∣<1C\left| \frac{1}{\left( x-3 \right) ^2} \right|>C\Leftrightarrow \left| x-3 \right|<\frac{1}{\sqrt{C}}

Thus for every C>0 there exists Ξ΄=1C\delta =\frac{1}{\sqrt{C}} such that ∣xβˆ’3∣<Ξ΄β‡’βˆ£1(xβˆ’3)2∣>C\left| x-3 \right|<\delta \Rightarrow \left| \frac{1}{\left( x-3 \right) ^2} \right|>C

By the definition

lim⁑xβ†’31(xβˆ’3)2=∞\underset{x\rightarrow 3}{\lim}\frac{1}{\left( x-3 \right) ^2}=\infty


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