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Hi, I have my exam tomorrow and have been doing the practice questions. However we dont get the answers to these questions so I am lost as to whether I am doing them right, also I am stuck at a few points, particularly with the definition questions.

\[

1.)Prove\ lim_{x \to \infty}\frac{2n+3}{n+1}=2 \\

I\ did:\ |a_n-a_m|=|(a_n-a)-(a_m-a)|\leq|a_n-a|+|a_m-a| \\

=|\frac{2n+3}{n+1}-2|+|\frac{2m+3}{m+1}-2|=|\frac{1}{n+1}|+|\frac{1}{m+1}|\leq\frac{1}{n}+\frac{1}{m}=\frac{m+n}{mn}\\

\text{not sure how to get that less than epsilon. usually i have m-n so can take out the m's.}\\

\ \\

\ \\

2.)Prove\ by\ definition\ f(x)=|x|\ is\ continuous\ at\ x_0=0\\

I\ have\ the\ definition\ as\ \forall\epsilon>0\ \exists\delta>0\ such\ that\ \forall y\in\mathbb{R}\ |x_0-y|<\delta\ implies\ |f(x_0)-f(y)|<\epsilon"\\

so\ I\ have:\ |0-y|=|-y|<\delta\ \rightarrow\ ||0|-|y||=|-y|<\epsilon\\

I\ know\ I'm\ supposed\ to\ relate\ delta\ and\ epsilon\ together\ somehow\ but\ cant\ for\ the\ life\ of\ me\ figure\ out\ why.\ The\ notes\ we\ have\ are\ very\ confusing\\

\]

Thank you for the help. I would be so lost without this forum

Carla x

\[

1.)Prove\ lim_{x \to \infty}\frac{2n+3}{n+1}=2 \\

I\ did:\ |a_n-a_m|=|(a_n-a)-(a_m-a)|\leq|a_n-a|+|a_m-a| \\

=|\frac{2n+3}{n+1}-2|+|\frac{2m+3}{m+1}-2|=|\frac{1}{n+1}|+|\frac{1}{m+1}|\leq\frac{1}{n}+\frac{1}{m}=\frac{m+n}{mn}\\

\text{not sure how to get that less than epsilon. usually i have m-n so can take out the m's.}\\

\ \\

\ \\

2.)Prove\ by\ definition\ f(x)=|x|\ is\ continuous\ at\ x_0=0\\

I\ have\ the\ definition\ as\ \forall\epsilon>0\ \exists\delta>0\ such\ that\ \forall y\in\mathbb{R}\ |x_0-y|<\delta\ implies\ |f(x_0)-f(y)|<\epsilon"\\

so\ I\ have:\ |0-y|=|-y|<\delta\ \rightarrow\ ||0|-|y||=|-y|<\epsilon\\

I\ know\ I'm\ supposed\ to\ relate\ delta\ and\ epsilon\ together\ somehow\ but\ cant\ for\ the\ life\ of\ me\ figure\ out\ why.\ The\ notes\ we\ have\ are\ very\ confusing\\

\]

Thank you for the help. I would be so lost without this forum

Carla x

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