Step 1: Identify the limit we want to calculate: lim (x -> ∞) (5x^2 + 3)/(2x^2 + 1).
Step 2: Notice that both the numerator and denominator have the highest power of x as x^2.
Step 3: To simplify, divide every term in the numerator and denominator by x^2.
Step 4: Rewrite the expression: (5x^2/x^2 + 3/x^2)/(2x^2/x^2 + 1/x^2).
Step 5: This simplifies to (5 + 3/x^2)/(2 + 1/x^2).
Step 6: Now, take the limit as x approaches infinity. As x becomes very large, 3/x^2 and 1/x^2 approach 0.
Step 7: So, the limit becomes (5 + 0)/(2 + 0) = 5/2.
Limits at Infinity – Understanding how to evaluate limits as the variable approaches infinity, particularly for rational functions.
Dominant Terms – Identifying the dominant terms in the numerator and denominator when calculating limits at infinity.
Simplifying Rational Functions – Using algebraic manipulation, such as dividing by the highest power of x, to simplify expressions for limit evaluation.
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