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

Question

BlackTom AI · Accepted Answer

The question asks to solve the limit as x approaches infinity of the rational expression (2x^2 + 4x - 6) / (-2x^2 - 7x + 10) and to write the answer to three decimal places.

A robust way to handle limits at infinity for rational functions is to divide every term by the highest power of x in the denominator, which here is x^2. Doing so, we rewrite the expression as:

(2x^2 + 4x - 6) / (-2x^2 - 7x + 10) = [ (2x^2)/x^2 + (4x)/x^2 + (-6)/x^2 ] / [ (-2x^2)/x^2 + (-7x)/x^2 + 10/x^2 ]
= (2 + 4/x - 6/x^2) / (-2 - 7/x + 10/x^2).

Now examine the limit as x → ∞. Each term with a factor 1/x or 1/x^2 tends to 0:
- 4/x → 0, -6/x^2 → 0, -7/x → 0, 10/x^2 → 0.
Thus the numerator tends to 2 and the denominator tends to -2, giving a limit of 2 / (-2) = -1.

Therefore, the limit is -1.000 when written to three decimal places.

If there were multiple choices, we would compare them to this result: any option claiming the limit equals -1.000 would be correct, while others would be incorrect because they would imply a different leading behavior (e.g., 0, 1, or divergent values) based on flawed division or misidentification of dominant terms.

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Compute [math]\displaystyle \lim _{x\to -\infty }{\frac{x^3}{e^{-3x}}}. (Use I to stand for [math]\infty if needed.)

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