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→∞(−2x+7x2−3x2+−4)] \frac{lim}{x \rightarrow \infty } (\frac{-2x+7x^2}{-3x^2+-4})

Question

BlackTom AI · Accepted Answer

We are asked to evaluate the limit as x approaches infinity of the rational expression (-2x + 7x^2) / (-3x^2 - 4).

A common technique for limits at infinity is to divide every term by the highest power of x in the expression, here x^2.

Divide numerator and denominator by x^2: 
- Numerator: (-2x)/x^2 + (7x^2)/x^2 = (-2/x) + 7.
- Denominator: (-3x^2)/x^2 + (-4)/x^2 = -3 + (-4/x^2).

So the expression becomes [(-2/x) + 7] / [-3 + (-4/x^2)].

Now take the limit as x → ∞:
- The term (-2/x) → 0 because it is proportional to 1/x.
- The term (-4/x^2) → 0 because it is proportional to 1/x^2.

Thus the expression tends to 7 / (-3) = -7/3.

If you prefer a quick leading-term argument: the degrees of the numerator and denominator are both 2, so the limit is the ratio of the leading coefficients, which are 7 in the numerator and -3 in the denominator, giving -7/3.

Finally, converting to a decimal for three decimal places would yield approximately -2.333, matching the value obtained by the reasoning above.

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