"Find the limit: limx→∞3x6−x4+2x2−34x5+2x3−5" was answered by students like you

jernplate8

Answered question

2021-01-02

Find the limit:

lim_{x \to \infty} \frac{3 x^{6} - x^{4} + 2 x^{2} - 3}{4 x^{5} + 2 x^{3} - 5}

Tuthornt

Skilled2021-01-03Added 107 answers

lim_{x \to \infty} \frac{3 x^{6} - x^{4} + 2 x^{2} - 3}{4 x^{5} + 2 x^{3} - 5}

Multiply and divided by

x^{5}

lim_{x \to \infty} \frac{3 x^{6} - x^{4} + 2 x^{2} - 3}{4 x^{5} + 2 x^{3} - 5} = lim_{x \to \infty} \frac{x^{5} \frac{3 x^{6} - x^{4} + 2 x^{2} - 3}{x^{5}}}{x^{5} \frac{4 x^{5} + 2 x^{3} - 5}{x^{5}}}

Divide:

lim_{x \to \infty} \frac{x^{5} \frac{3 x^{6} - x^{4} + 2 x^{2} - 3}{x^{5}}}{x^{5} \frac{4 x^{5} + 2 x^{3} - 5}{x^{5}}} = lim_{x \to \infty} \frac{3 x - \frac{1}{x} + \frac{2}{x^{3}} - \frac{3}{x^{5}}}{4 + \frac{2}{x^{2}} - \frac{5}{x^{5}}}

The limit of the quotient is the quotient of limits:

lim_{x \to \infty} \frac{3 x - \frac{1}{x} + \frac{2}{x^{3}} - \frac{3}{x^{5}}}{4 + \frac{2}{x^{2}} - \frac{5}{x^{5}}} = \frac{lim_{x \to \infty} (3 x - \frac{1}{x} + \frac{2}{x^{3}} - \frac{3}{x^{5}})}{lim_{x \to \infty} (4 + \frac{2}{x^{2}} - \frac{5}{x^{5}})}

The limit of a sum/difference is the sum/difference of limits:

\frac{lim_{x \to \infty} (3 x - \frac{1}{x} + \frac{2}{x^{3}} - \frac{3}{x^{5}})}{lim_{x \to \infty} (4 + \frac{2}{x^{2}} - \frac{5}{x^{5}})} = \frac{(- lim_{x \to \infty} \frac{3}{x^{5}} + lim_{x \to \infty} \frac{2}{x^{3}} - lim_{x \to \infty} \frac{1}{x} + lim_{x \to \infty} 3 x)}{lim_{x \to \infty} (4 + \frac{2}{x^{2}} - \frac{5}{x^{5}})}

The limit of a quotient is the quotient of limits:

\frac{(- lim_{x \to \infty} \frac{3}{x^{5}} + lim_{x \to \infty} \frac{2}{x^{3}} - lim_{x \to \infty} \frac{1}{x} + lim_{x \to \infty} 3 x)}{lim_{x \to \infty} (4 + \frac{2}{x^{2}} - \frac{5}{x^{5}})} = \frac{(- lim_{x \to \infty} \frac{3}{x^{5}} + lim_{x \to \infty} \frac{2}{x^{3}} - lim_{x \to \infty} 3 x) - \frac{lim_{x \to \infty} 1}{lim_{x \to \infty} x}}{lim_{x \to \infty} (4 + \frac{2}{x^{2}} - \frac{5}{x^{5}})}

The limit of a constant is equal to the constant:

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