Questions: Divide the numerator and the denominator by the highest power of (x) in the denominator and proceed from there. Find the limit. Write (infty) or (-infty) where appropriate. (lim x rightarrow-infty fracsqrt[3]x-3 x+52 x+x^2/3-7) (lim x rightarrow-infty fracsqrt[3]x-3 x+52 x+x^2/3-7=)

Solution Steps

To find the limit as \( x \) approaches \(-\infty\) for the given expression, we need to divide both the numerator and the denominator by the highest power of \( x \) in the denominator. This will help us simplify the expression and determine the limit.

1. Identify the highest power of \( x \) in the denominator, which is \( x \).
2. Divide every term in both the numerator and the denominator by \( x \).
3. Simplify the resulting expression.
4. Evaluate the limit as \( x \) approaches \(-\infty\).

The highest power of \( x \) in the denominator is \( x \).

Divide every term in both the numerator and the denominator by \( x \): \[ \frac{\sqrt[3]{x} - 3x + 5}{2x + x^{\frac{2}{3}} - 7} \div x = \frac{\frac{\sqrt[3]{x}}{x} - \frac{3x}{x} + \frac{5}{x}}{\frac{2x}{x} + \frac{x^{\frac{2}{3}}}{x} - \frac{7}{x}} \]

Simplify the terms: \[ \frac{\frac{x^{\frac{1}{3}}}{x} - 3 + \frac{5}{x}}{2 + \frac{x^{\frac{2}{3}}}{x} - \frac{7}{x}} = \frac{x^{-\frac{2}{3}} - 3 + \frac{5}{x}}{2 + x^{-\frac{1}{3}} - \frac{7}{x}} \]

As \( x \to -\infty \):

• \( x^{-\frac{2}{3}} \to 0 \)
• \( \frac{5}{x} \to 0 \)
• \( x^{-\frac{1}{3}} \to 0 \)

Thus, the expression simplifies to: \[ \frac{0 - 3 + 0}{2 + 0 - 0} = \frac{-3}{2} \]

Final Answer