Question

Can someone show me how to evaluate limits at infinity and negative infinity?

Answer 1

For example, \[\lim _{x \rightarrow \infty} \frac{-2x^3+x^2-3 }{ x^2+4x+7 }\]

Answer 2

Okay so first...when it comes to infinity (and negative infinity) you need only focus on the highest degree in both the denominator and numerator

why? well because those are the numbers that grow the fastest as your numbers get bigger.

Answer 3

So really, all you need to look at is

\[\large \lim_{x\rightarrow \infty} \frac{-2x^3}{x^2}\]

Answer 4

Now lets do something...lets plug in 10 for 'x'

\[\large \frac{-2(10)^3}{10^2} = \frac{-2000}{100} = -20\]
Now what about 100?
\[\large \frac{-2(100)^3}{100^2} = \frac{-2000000}{10000} = -20000\]

What about 1000? and 10000? etc...see whats happening? as x gets bigger and bigger...we approach bigger and bigger numbers...also known as approaching infinity.

Answer 5

In general, after noticing what happens there we can look at other examples...what about if the -2x^3 was in the denominator and the x^2 was in the numerator?

\[\large \lim_{x \rightarrow \infty} \frac{x^2}{-2x^3}\]

if you put bigger and bigger numbers in there, what would this approach?