Find the limit as x approaches negative infinity. lim_(x rarr-oo)(5x^(3)-3x)/(sqrt(4x^(6)-7))=

Identify highest power of x: Identify the highest power of x in both the numerator and the denominator. In the numerator, the highest power of x is x^3. In the denominator, the highest power of x inside the square root is x^6, which becomes x^3 when taken outside the square root.Divide by highest power: Divide both the numerator and the denominator by x^3, the highest power of x found in the previous step. lim_(x rarr-oo) (5x^3/x^3 - 3x/x^3) / (sqrt(4x^6/x^3 - 7/x^3))Simplify expression: Simplify the expression by canceling out the x^3 terms and reducing the fractions. lim_(x rarr-oo) (5 - 3/x^2) / (sqrt(4 - 7/x^6))Approach negative infinity: As x approaches negative infinity, the terms with x in the denominator approach zero. lim_(x rarr-oo) (5 - 0) / (sqrt(4 - 0))Evaluate limit: Evaluate the limit by substituting the values that approach zero. lim_(x rarr-oo) 5 / sqrt(4) = 5 / 2

Home

Math Problems

Algebra 1

Powers with decimal and fractional bases

Full solution

Q. Find the limit as

x

approaches negative infinity.

\newline

\lim _{x \rightarrow-\infty} \frac{5 x^{3}-3 x}{\sqrt{4 x^{6}-7}}=

Identify highest power of x: Identify the highest power of $x$ in both the numerator and the denominator. $\newline$ In the numerator, the highest power of $x$ is $x^3$ . In the denominator, the highest power of $x$ inside the square root is $x^6$ , which becomes $x^3$ when taken outside the square root.
Divide by highest power: Divide both the numerator and the denominator by $x^3$ , the highest power of $x$ found in the previous step. $\newline$ $\lim_{x \rightarrow -\infty} \frac{\frac{5x^3}{x^3} - \frac{3x}{x^3}}{\sqrt{\frac{4x^6}{x^3} - \frac{7}{x^3}}}$
Simplify expression: Simplify the expression by canceling out the $x^3$ terms and reducing the fractions. $\lim_{x \to -\infty} \frac{5 - \frac{3}{x^2}}{\sqrt{4 - \frac{7}{x^6}}}$
Approach negative infinity: As $x$ approaches negative infinity, the terms with $x$ in the denominator approach zero. $\lim_{x \rightarrow -\infty} \frac{(5 - 0)}{(\sqrt{4 - 0})}$
Evaluate limit: Evaluate the limit by substituting the values that approach zero. $\newline$ $\lim_{x \rightarrow -\infty} \frac{5}{\sqrt{4}} = \frac{5}{2}$