Step 1: Identify the limit we want to find: lim (x -> ∞) (3x^2 + 2)/(5x^2 - 4x + 1).
Step 2: Look at the highest power of x in the numerator and denominator. The highest power in both is x^2.
Step 3: Rewrite the limit by focusing on the leading terms (the terms with the highest power of x). This gives us: lim (x -> ∞) (3x^2)/(5x^2).
Step 4: Simplify the expression (3x^2)/(5x^2) by canceling x^2 from the numerator and denominator. This results in: 3/5.
Step 5: Conclude that as x approaches infinity, the limit is 3/5.
Limits at Infinity – Understanding how to evaluate limits as the variable approaches infinity, particularly focusing on the leading terms in polynomial expressions.
Dominance of Leading Terms – Recognizing that in rational functions, the highest degree terms in the numerator and denominator will dominate the behavior of the function as x approaches infinity.
