Find the limit.
lim_(x->infinity)(sqrt(9 x^2 + x) - 3 x)
To find the limit as x approaches infinity for the given function, we need to simplify the expression and see what happens as x becomes larger and larger.
Let's start by simplifying the expression inside the square root:
sqrt(9x^2 + x) - 3x
Squaring both sides (to eliminate the square root) gives us:
(9x^2 + x) - 6x(sqrt(9x^2 + x)) + 9x^2
Combining like terms, we have:
9x^2 + x - 6x(sqrt(9x^2 + x)) + 9x^2
Next, we can factor out an x from the remaining terms:
x(18x + 1 - 6(sqrt(9x^2 + x)))
Now, we can see that as x approaches infinity, the dominant term will be 18x^2. The other terms become relatively insignificant. Therefore, the limit as x approaches infinity is:
lim_(x->infinity)(x(18x + 1 - 6(sqrt(9x^2 + x))))
= lim_(x->infinity)(18x^2)
Since the coefficient (18) is nonzero, the limit evaluates to positive infinity.
Therefore, the limit as x approaches infinity for the given function is positive infinity.