Question

How do you find the limit of `(8x-14)/(sqrt(13x+49x^2))` as x approaches `oo`?

Answer 1

Do a little factoring and canceling to get `lim_(x->oo)(8x-14)/(sqrt(13x+49x^2))=8/7`.

Explanation:

At limits of infinity, the general strategy is to take advantage of the fact that `lim_(x->oo)1/x=0`. Normally that means factoring out an `x`, which is what we'll be doing here.

Begin by factoring an `x` out of the numerator and an `x^2` out of the denominator:
`(x(8-14/x))/(sqrt(x^2(13/x+49)))`
`=(x(8-14/x))/(sqrt(x^2)sqrt(13/x+49))`

The issue is now with `sqrt(x^2)`. It is equivalent to `abs(x)`, which is a piecewise function:
`abs(x)={(x, "for",x > 0),(-x,"for",x < 0):}`

Since this is a limit at positive infinity (`x>0`), we will replace `sqrt(x^2)` with `x`:
`=(x(8-14/x))/(xsqrt(13/x+49))`

Now we can cancel the `x`s:
`=(8-14/x)/(sqrt(13/x+49))`

And finally see what happens as `x` goes to `oo`:
`=(8-14/oo)/(sqrt(13/oo+49))`

Because `lim_(x->oo)1/x=0`, this is equal to:
`(8-0)/(sqrt(0+49))`
`=8/sqrt(49)`
`=8/7`