Question

Find the limit if it exists
lim as x-> infinity 4/(x-3)^3

Answer 1

What do you think the answer is?

Answer 2

I already know the answer- 0, but I don't understand how. Limits at infinity get me lost.

Answer 3

Okay think about it like this

Since we want to know the limit as we approach infinity...aka REALLY BIG NUMBERS!!! lets say I plug in 1000 for 'x'

\[\large \frac{4}{(1000 - 3)^3}\] That will come out to a REALLY small number right?
What about if I plug in 10000 now?

\[\large \frac{4}{(10000 - 3)^3}\]

That would get even smaller still right?

As values of 'x' get larger and larger approaching infinity....the limit gets smaller and smaller approaching zero

Answer 4

okay i understand this but now im confused between
limits at infinity and infinite limits :(

Answer 5

Not sure what you mean, do you have an example?

Answer 6

limit as x approaches infinity vs limit as x approaches a number(from + or -)

Answer 7

limas x->-1+ 2/(x+1) answer is infinity

Answer 8

and in the previous answer i asked you the answer was 0

Answer 9

Sorry for the extremely late response!

Lets work with that example you gave

\[\large lim_{x \rightarrow -1^+} \frac{2}{(x + 1)}\]

What would happen if you plug in -1 into that equation for 'x'?

\[\large \frac{2}{(-1 + 1)} = \frac{-2}{0}\]

What do we know about dividing by 0? The answer is infinite

Why? think about it...if I plug lets say 1 in for x in the equation \(\large \frac{1}{x}\) the answer is 1...what about if I plug in 1/2? the answer would be 2

What about 1/4? the answer would be 4

What about 1/1000? The answer would be 1000 etc... as the numbers get smaller and smaller *closer to 0* the value of the expression gets bigger and bigger *approaching infinity*

Answer 10

Thank you so much! You're awesome :)

Answer 11

It's zero. Just a starting tip. You should consider both the situations when x approaches positive and negative infinity.