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.

类似问题

Compute \(\displaystyle \lim _{x\to -\infty }{\frac {x^4-8x^3-3x+5}{6x^4+7x^2+3}}\). (Use I to stand for \(\infty \) if needed.)

Compute [math: limx→−∞x4−8x3−3x+56x4+7x2+3]\displaystyle \lim _{x\to -\infty }{\frac {x^4-8x^3-3x+5}{6x^4+7x^2+3}}. (Use I to stand for [math: ∞]\infty if needed.)

Compute [math: limx→−∞−4x3+2x2+6x−57x4+4x2+1]\displaystyle \lim _{x\to -\infty }{\frac {-4x^3+2x^2+6x-5}{7x^4+4x^2+1}}. (Use I to stand for [math: ∞]\infty if needed.)

Compute [math: limx→−∞x7]\displaystyle \lim _{x\to -\infty }{x^7}. (Use I to stand for [math: ∞]\infty if needed.)

Compute \(\displaystyle \lim _{x\to -\infty }{\frac {3x^4-4x^3-2x+1}{8x^4+9x^2+5}}\). (Use I to stand for \(\infty \) if needed.)

Compute [math: limx→−∞3x4−4x3−2x+18x4+9x2+5]\displaystyle \lim _{x\to -\infty }{\frac {3x^4-4x^3-2x+1}{8x^4+9x^2+5}}. (Use I to stand for [math: ∞]\infty if needed.)

Compute limx→−∞x13[math]\displaystyle \lim _{x\to -\infty }{x^{\frac {1}{3}}}. (Use I to stand for ∞[math]\infty if needed.)

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