Question

someone help me solve these two questions
1) Prove that Lim x-->-1- 5/(x+1)^3=-infinity

2)prove using definition 8, that lim x->infinity x^3=infinity

Answer 1

let L=
\[\lim_{x \rightarrow -1-}\frac{ -5 }{ \left( x+1 \right)^3 }\]
put x=-1-h,h>0,
\[h \rightarrow0~as~x \rightarrow-1-\]
\[L=\lim_{h \rightarrow 0}\frac{ 5 }{ \left( -1-h +1\right)^3 }=\lim_{h \rightarrow 0}\frac{ -5 }{ h^3 }\rightarrow -\infty \]

Answer 2

Your definitions may vary, but here are the ones I'll be referring to:

One-sided limits:
\[\large\lim_{x\to c^-}f(x)=L\]
if for any \(\epsilon>0\), there exists \(\delta>0\) such that
\[\large-\delta<x-c<0~~\implies~~|f(x)-L|<\epsilon\]
Infinite limits:
\[\large\lim_{x\to c}f(x)=-\infty\]
if for any \(N<0\) there exists \(\delta>0\) such that
\[\large0<|x-c|<\delta~~\implies~~f(x)<N\]
For your particular limit,
\[\large\lim_{x\to-1^-}\frac{5}{(x+1)^3}=-\infty\]
you have to show that for any \(M<0\) there is some \(\delta>0\) such that
\[\large-\delta<|x+1|<0~~\implies~~\frac{5}{(x+1)^3}<M\]

Answer 3

Oops, that last line should read
\[\large -\delta<x+1<0~~\implies\cdots\]