NOTE: Due to formatting constraints, limits are shown with a horizontal bar in these quiz questions.For example: will be written as: [math: limx→af(x)] \frac{lim}{x \rightarrow a }f(x) Solve the following limit and write your answer to three decimal places.[math: limx→∞(2x2+4x+−6−2x2+−7x+10)] \frac{lim}{x \rightarrow \infty } (\frac{2x^2+4x+-6}{-2x^2+-7x+10})

Question

BlackTom AI · Accepted Answer

We are asked to evaluate a limit of a rational function as x approaches infinity.

First, identify the highest power of x in both the numerator and the denominator. In the numerator, the leading term is 2x^2; in the denominator, the leading term is -2x^2.

A standard technique for limits at infinity is to divide every term by the highest power of x present, here x^2, to simplify the expression and reveal the behavior as x grows without bound.

Divide each term by x^2:
- Numerator: (2x^2 + 4x - 6)/x^2 = 2 + 4/x - 6/x^2
- Denominator: (-2x^2 - 7x + 10)/x^2 = -2 - 7/x + 10/x^2

Thus the limit becomes lim_{x->∞} [ (2 + 4/x - 6/x^2) / (-2 - 7/x + 10/x^2) ].

As x → ∞, the terms containing 1/x and 1/x^2 vanish, leaving the limit of the dominant constant terms: 2 / (-2).

Therefore, the expression tends to -1 in the infinite limit.

To three decimal places, this value is -1.000.

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