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lim x->inf (3x5 - 9x + 2)/(2x2 + 5x + 1)

Mathematics

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Answer

∞\infty

Step-by-step solution

4 steps

  1. Identify the highest power of xx in the denominator.

    The given expression is a rational function. To evaluate the limit as x→∞x \to \infty, we divide every term in the numerator and the denominator by the highest power of xx in the denominator.
    The denominator is 2x2+5x+12x^2 + 5x + 1, and its highest power of xx is x2x^2.

  2. Divide each term by x2x^2.

    lim⁡x→∞(3x5x2−9xx2+2x22x2x2+5xx2+1x2)\lim_{x \to \infty} \left(\frac{\frac{3x^5}{x^2} - \frac{9x}{x^2} + \frac{2}{x^2}}{\frac{2x^2}{x^2} + \frac{5x}{x^2} + \frac{1}{x^2}}\right)

  3. Simplify the expression.

    lim⁡x→∞(3x3−9x+2x22+5x+1x2)\lim_{x \to \infty} \left(\frac{3x^3 - \frac{9}{x} + \frac{2}{x^2}}{2 + \frac{5}{x} + \frac{1}{x^2}}\right)

  4. Evaluate the limit by substituting x→∞x \to \infty.

    As x→∞x \to \infty, terms of the form cxn\frac{c}{x^n} (where cc is a constant and n>0n > 0) approach 0.

    3(∞)3−0+02+0+0\frac{3(\infty)^3 - 0 + 0}{2 + 0 + 0} ∞2\frac{\infty}{2} ∞\infty
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