Calculate the limit: lim (x -> ∞) (3x^2 + 2)/(5x^2 - 4) (2023)

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Question: Calculate the limit: lim (x -> ∞) (3x^2 + 2)/(5x^2 - 4) (2023)

Options:

Correct Answer: 3/5

Exam Year: 2023

Solution:

Dividing numerator and denominator by x^2 gives lim (x -> ∞) (3 + 2/x^2)/(5 - 4/x^2) = 3/5.

Calculate the limit: lim (x -> ∞) (3x^2 + 2)/(5x^2 - 4) (2023)

Calculate the limit: lim (x -> ∞) (3x^2 + 2)/(5x^2 - 4) (2023)

Step 1: Identify the limit we want to calculate: lim (x -> ∞) (3x^2 + 2)/(5x^2 - 4).
Step 2: Notice that both the numerator (3x^2 + 2) and the denominator (5x^2 - 4) are polynomials of degree 2.
Step 3: To simplify, divide every term in the numerator and the denominator by x^2, the highest power of x in the expression.
Step 4: Rewrite the expression: (3x^2/x^2 + 2/x^2)/(5x^2/x^2 - 4/x^2).
Step 5: This simplifies to (3 + 2/x^2)/(5 - 4/x^2).
Step 6: Now, take the limit as x approaches infinity. As x becomes very large, 2/x^2 approaches 0 and 4/x^2 approaches 0.
Step 7: Therefore, the limit simplifies to (3 + 0)/(5 - 0) = 3/5.
Step 8: Conclude that the limit is 3/5.

Limits at Infinity – Understanding how to evaluate limits as the variable approaches infinity, particularly for rational functions.
Rational Functions – Analyzing the behavior of polynomials in the numerator and denominator as they grow large.
Dominant Terms – Identifying the leading terms in the numerator and denominator that dictate the limit's value.