Limits at Infinity, Part I — Question 1

Question 1

Evaluate the limit: $$\lim_{x \to \infty} \frac{6x^3 - 4x^2 + 10}{3x^3 + 5x - 7}$$

Question 1 - Solution

We are given: $$\lim_{x \to \infty} \frac{6x^3 - 4x^2 + 10}{3x^3 + 5x - 7}$$

Step 1: Identify the dominant terms.

As $x \to \infty$, the highest degree terms dominate: Numerator: $6x^3$ , Denominator: $3x^3$

So we focus on: $$\frac{6x^3}{3x^3} = 2$$

Step 2: Divide all terms by $x^3$: $$\lim_{x \to \infty} \frac{6 - \frac{4}{x} + \frac{10}{x^3}}{3 + \frac{5}{x^2} - \frac{7}{x^3}} = \frac{6}{3} = 2$$

Conclusion: $$\boxed{\lim_{x \to \infty} \frac{6x^3 - 4x^2 + 10}{3x^3 + 5x - 7} = 2}$$