Find the limit as x approaches infinity.

sqrt(81x^2 + x)− 9x.

I've worked this two different ways and have gotten 0 and 1/9, neither of which is the right answer according to the homework system??

√(81x^2 + x)− 9x

= (√(81x^2 + x)− 9x)(√(81x^2 + x)+ 9x)/(√(81x^2 + x)+ 9x)

= ((81x^2+x)-81x^2)/(√(81x^2 + x)+ 9x)
= x/(√(81x^2 + x)+ 9x)
= 1/(√(81 + 1/x)+ 9)
as x->∞, that becomes
1/(√81+9) = 1/18

Well, let's solve this puzzle together, but I warn you, it might get a bit "square"y.

As x approaches infinity, we can disregard the x in the expression because compared to 81x^2, it's quite a tiny fella. So we are left with sqrt(81x^2) − 9x.

Now let's simplify the first square root term. Sqrt(81x^2) is just 9x because sqrt(81) is 9. So we have 9x - 9x.

And what do we have after this frenzied subtraction? A big fat 0! So the limit as x approaches infinity of our expression is indeed 0.

Yup, it looks like the homework system got a bit "radical" this time. Remember, math problems can often be quite tricky, but never as tricky as a clown trying to juggle with eggs.

To find the limit as x approaches infinity of the function sqrt(81x^2 + x) - 9x, we can simplify it by dividing through by x.

Let's rewrite the function:
f(x) = sqrt(81x^2 + x) - 9x

Divide through by x:
f(x) = (sqrt(81x^2 + x) - 9x) / x

Now, we'll simplify the expression:
f(x) = (sqrt(81x^2 + x) / x) - 9

As x approaches infinity, the value of sqrt(81x^2 + x) / x tends to 81, because the highest power of x is in the numerator. Thus, we have:
f(x) = 81 - 9
f(x) = 72

Therefore, the limit as x approaches infinity of sqrt(81x^2 + x) - 9x is 72.

To find the limit as x approaches infinity for the function sqrt(81x^2 + x) - 9x, we can use algebraic manipulation and limits properties. Let's go through the steps to solve it correctly.

First, let's simplify the expression:
sqrt(81x^2 + x) - 9x

Next, we can divide the entire expression by x:
(sqrt(81x^2 + x) - 9x) / x

Now, we want to find the limit as x approaches infinity. We can rewrite the expression as:
(sqrt(81x^2 + x) / x) - 9

As x approaches infinity, the term (81x^2 + x) becomes dominated by the x^2 term. So we can simplify further by dividing both the numerator and denominator by x:
(sqrt(81 + (x/x^2)) - 9)

Since x is going to infinity, x/x^2 tends to zero. Therefore, we have:
(sqrt(81 + 0) - 9)

Now, it simplifies to:
sqrt(81) - 9

Finally, we have:
9 - 9
0

So, the limit of sqrt(81x^2 + x) - 9x as x approaches infinity is 0.

If you obtained a different answer, it's possible that there might have been a mistake made during the simplification or calculation process.