Questions: Evaluate the limit using the appropriate properties of limits. lim x → ∞ (5x^2 - x + 6) / (2x^2 + 5x - 4)

Transcript text: Evaluate the limit using the appropriate properties of limits. \[ \lim _{x \rightarrow \infty} \frac{5 x^{2}-x+6}{2 x^{2}+5 x-4} \]

Solution

Solution Steps

To evaluate the limit as \( x \) approaches infinity for the given rational function, we can use the properties of limits and focus on the highest degree terms in the numerator and the denominator. The highest degree terms will dominate the behavior of the function as \( x \) approaches infinity.

Solution Approach

Identify the highest degree terms in the numerator and the denominator.
Simplify the expression by dividing both the numerator and the denominator by \( x^2 \).
Evaluate the limit of the simplified expression as \( x \) approaches infinity.

Step 1: Identify the Highest Degree Terms

To evaluate the limit as \( x \) approaches infinity, we first identify the highest degree terms in both the numerator and the denominator. The highest degree term in the numerator is \( 5x^2 \) and in the denominator is \( 2x^2 \).

Step 2: Simplify the Expression

We simplify the expression by dividing both the numerator and the denominator by \( x^2 \):

\[ \frac{5x^2 - x + 6}{2x^2 + 5x - 4} = \frac{5 - \frac{1}{x} + \frac{6}{x^2}}{2 + \frac{5}{x} - \frac{4}{x^2}} \]

Step 3: Evaluate the Limit

As \( x \) approaches infinity, the terms \(\frac{1}{x}\), \(\frac{6}{x^2}\), \(\frac{5}{x}\), and \(\frac{4}{x^2}\) approach 0. Therefore, the expression simplifies to:

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