`lim_(x->oo)` `x^2/ sqrt(x^4+1) `

Expert Answers

An illustration of the letter 'A' in a speech bubbles

`lim_(x->oo)x^2/(sqrt((x^4+1)))=1`

(1) Checking numerically we find the following:

x |f(x)
-------
0 0
1 .70711
2 .97014
3 .99398
4 .99805
5 .9992
6 .99961

This seems to suggest 1 as a limit.

(2) The graph (attached) shows the function approaches 1 from below.

(3) Algebraically:

`lim_(x->oo) x^2/sqrt(x^4+1)`

`=lim_(x->oo) (x^2*1/sqrt(x^4))/(sqrt(x^4+1)*1/sqrt(x^4))`

`=lim_(x->oo) 1/sqrt(1+1/x^4)`

`=(lim_(x->oo)1)/(lim(x->oo)sqrt(1+1/x^4))`

`=1/(sqrt(1+0))` using `lim_(x->oo)1/x^c=0`

=1

Images:
This image has been Flagged as inappropriate Click to unflag
Image (1 of 1)
Last Updated by eNotes Editorial on