Question

Use the properties of limits to help decide whether each limit exists. If a limit exists, find its value. lim x→[infinity]

​
6x 2
−2x+1
3x 2
+2x
​
Select the correct choice below and, if necessary, fill in the answer box within your choice. A. lim x→[infinity]
​
6x 2
−2x+1
3x 2
+2x
​
= (Simplify your answer. Type an integer or a fraction.) B. The limit does not exist and is neither [infinity] nor −[infinity].

Answer 1

To determine the limit of the given expression as x approaches infinity, you can use the properties of limits, specifically the idea of comparing the degrees of the leading terms in the numerator and denominator.

In your expression:

lim(x→∞) (6x^2 - 2x + 1) / (3x^2 + 2x)

You can compare the degrees of the leading terms (the terms with the highest powers of x) in the numerator and denominator:

Numerator: 6x^2

Denominator: 3x^2

The degree of the leading term in the numerator is 2, and the degree of the leading term in the denominator is also 2. When the degrees are the same, you can compare the coefficients of these leading terms. In this case, the coefficient of 6x^2 in the numerator is 6, and the coefficient of 3x^2 in the denominator is 3.

Now, calculate the limit:

lim(x→∞) (6x^2 - 2x + 1) / (3x^2 + 2x)

Since the degrees and coefficients of the leading terms are the same, you can divide both the numerator and denominator by the highest power of x, which is x^2:

lim(x→∞) (6x^2/x^2 - 2x/x^2 + 1/x^2) / (3x^2/x^2 + 2x/x^2)

Now, simplify:

lim(x→∞) (6 - 2/x + 1/x^2) / (3 + 2/x)

As x approaches infinity, the terms with 1/x and 1/x^2 become negligible compared to other terms, as they go to zero. Therefore, you can ignore these terms:

lim(x→∞) (6 - 0 + 0) / (3 + 0)

Now, you have a limit with constants only:

lim(x→∞) (6 / 3) = 2

So, the limit as x approaches infinity is equal to 2. Therefore, the correct choice is:

A. lim(x→∞) (6x^2 - 2x + 1) / (3x^2 + 2x) = 2

Explanation: