Question

Find the limit of the function algebraically.

limit as x approaches zero of quantity x squared plus two x divided by x to the fourth power.

Answer 1

\[\lim_{x \rightarrow 0}\frac{ x ^{2} + 2x }{ x ^{4} }\]

Answer 2

it might help to look at a table of values. Specifically x values that are very close to 0

things like

x = -0.5
x = -0.4
x = -0.3
x = -0.2
x = -0.1
x = 0
x = 0.1
x = 0.2
x = 0.3
x = 0.4
x = 0.5

Answer 3

I'm assuming L'Hopitals wouldn't be valid in this situation?

Answer 4

You could use L'Hospitals' rule but only on that current form

if you were to simplify like so

\[\Large \frac{ x ^{2} + 2x }{ x ^{4} }\]
\[\Large \frac{ x(x + 2) }{ x ^{4} }\]
\[\Large \frac{ x(x + 2) }{ x*x ^{3} }\]
\[\Large \frac{ {\color{red}{x}}(x + 2) }{ {\color{red}{x}}*x ^{3} }\]
\[\Large \frac{ {\color{red}{\cancel{x}}}(x + 2) }{ {\color{red}{\cancel{x}}}*x ^{3} }\]
\[\Large \frac{ x+2 }{ x ^{3} }\]
then L'Hospitals' rule wouldn't work

Answer 5

In my opinion, it's always a good idea to simplify as much as possible before doing the limit

Answer 6

I agree 100%, I just wasn't sure if L'Hopitals was considered algebraically