"Solve the following limit. limx→∞(3x3−4x+27x3+5)" was answered by students like you

Carol Gates

Answered question

2021-10-11

Solve the following limit.

lim_{x \to \infty} (\frac{3 x^{3} - 4 x + 2}{7 x^{3} + 5})

izboknil3

Skilled2021-10-12Added 99 answers

Divide numerator and denominator of the expression with the highest power of x in the denominator.
Thus we can write the limit expression as,

lim_{x \to \infty} (\frac{3 x^{3} - 4 x + 2}{7 x^{3} + 5}) = lim_{x \to \infty} (\frac{3 \frac{x^{3}}{x^{3}} - 4 \frac{x}{x^{3}} + \frac{2}{x}}{7 \frac{x^{3}}{x^{3}} + \frac{5}{x^{3}}})

= lim_{x \to \infty} (\frac{3 - \frac{4}{x^{2}} + \frac{2}{x^{3}}}{7 + \frac{5}{x^{3}}})

Now simplify the limit expression using limit properties and apply the limit value to the variable x.
The terms,

lim_{x \to \infty} \frac{4}{x^{2}}, lim_{x \to \infty} \frac{2}{x^{3}}, lim_{x \to \infty} \frac{5}{x^{3}}

approaches 0 as x approaches infinity. In other words,

lim_{x \to \infty} \frac{4}{x^{2}} = \frac{4}{\infty}

= \frac{4}{\frac{1}{0}}

= 4 \times \frac{0}{1}

= 0

Similarly we get all the terms

lim_{x \to \infty} \frac{4}{x^{2}}, lim_{x \to \infty} \frac{2}{x^{3}}, lim_{x \to \infty} \frac{5}{x^{3}}

equal to 0.
Therefore we get,

lim_{x \to \infty} (\frac{3 - \frac{4}{x^{2}} + \frac{2}{x^{3}}}{7 + \frac{5}{x^{3}}}) = \frac{lim_{x \to \infty} (3 - \frac{4}{x^{2}} + \frac{2}{x^{3}})}{lim_{x \to \infty} (7 + \frac{5}{x^{3}})}

= \frac{3 - lim_{x \to \infty} \frac{4}{x^{2}} + lim_{x \to \infty} \frac{2}{x^{3}}}{7 + lim_{x \to \infty} \frac{5}{x^{3}}}

= \frac{3 - 0 + 0}{7 + 0} = \frac{3}{7}

Hence the limit of the given expression os

\frac{3}{7}

Jeffrey Jordon

Expert2022-06-29Added 2605 answers

Answer is given below (on video)

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