Question For the function ( f(x) ) given below, where ( x>-4...

Question For the function ( f(x) ) given below, where ( x>-4 ), evaluate ( lim _{x ightarrow infty} f(x) ). [ f(x)=ln (2 x+8)-ln left(6 x^{2}+7 ight) ] If the function increases without bound, you should enter ( infty ). If the function decreases without bound, you should enter ( -infty . ) If the function does not approach a finite limit nor ( pm infty ) as ( x ightarrow pm infty ), you should enter ( varnothing ). Provide your answer below: [ lim _{x ightarrow infty} f(x)= ]

Transcript

00:01 All right, so here we have this problem.

00:02 We want to take the limit as x goes to positive infinity of f of x.

00:06 And f of x is this function of these two logs being subtracted, natural logs being subtracted.

00:12 All right, so a couple things we need to realize here.

00:16 One, there's rules of logarithms that if you're subtracting two logs, two natural logs in this case, you can rewrite it as one log and as a fraction.

00:26 So really, f of x is equal to one natural log of 2x plus 8 on top and 6x squared plus 7 on the bottom.

00:49 Now, it would be nice if we could factor and reduce things down, but we can't.

00:52 But now we could look at this and go, hey, as i put infinity for x into f of x here, because the denominator is being squared has a bigger degree, that means that the limit of the f of x, as x goes to infinity of f of x, of the parentheses part, is going to equal basically the natural log of zero.

01:22 Now we don't really ever get to zero, but it's approaching zero.

01:26 So we got now got to think that maybe i should rewrite it like this.

01:33 So as we take the limit, not of f of x, but of that fraction, this fraction here, 2x plus 8 over 6x squared plus 7, that that's going to basically be zero because it's a bigger degree in the denominator.

01:51 All right.

01:52 Now, we can't evaluate the natural log of zero.

01:55 You can't take the log of zero.

01:57 But let's think what does y equal natural log of x? what does that graph look like? so here's my graph.

02:08 Think about it.

02:09 So it has a vertical asymptote at zero because we can't.

02:14 So we have a vertical asymptote right here at zero.

02:19 And we know that when we put zero in, one in for x, we get zero...

Question

Please give Ace some feedback