infinite limits: understanding how to calculate them

[annajee]

Can someone please help me understand how to calculate infinite limits? It's easy to see on a graph. And I think I understand the concept, but when I'm just looking at an equation it is super hard for me to figure out positive or negative infinity. For example: The left hand limit (as x approaches 4) of 3 / (4 - x)^3. Is it just me or does that take a whole lot of effort to figure it out. Is this actually difficult? somebody tell me the secret to these infinite limit problems.

Thanks.

Anna

 

[skeeter]

Re: infinite limits

edit ... fix my dyslexic post

no secret ... just common sense.

\(\displaystyle \lim_{x \to 4^-} \frac{3}{(4-x)^3}\)

pick a value close to 4 on the correct side ... say 3.9

you get \(\displaystyle \frac{3}{(.1)^3}\)

note the denominator stays positive and gets smaller as x gets closer to 4 ...

\(\displaystyle \frac{fixed \, positive \, value}{very \, small \, positive \, value} = very \, large \, positive \, value\)

the function value grows without bound toward positive infinity.

 

[Deleted member 4993]

Re: infinite limits

That is why graphing calculators were invented!!!

Anyway, since you are looking at behavior around x = 4, for approaching from the left

assume,

\(\displaystyle x \, = \, 4 \, - \, \epsilon\)

then

\(\displaystyle \frac{3}{(4-x)^3} \, = \, \frac{3}{(4-[4 \, - \, \epsilon])^3} \, = \, \frac{3}{ \epsilon^3}\)

so you see, from the left as you approach x = 4 - you approach +infinity.

Similarly for approaching from the left,

assume,

\(\displaystyle x \, = \, 4 \, + \, \epsilon\)

and do the same analysis (you will find that the function will approach -infinity)