Evaluate the limit: lim (x -> ∞) (3x^2 + 2)/(5x^2 - 4x + 1)

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Question: Evaluate the limit: lim (x -> ∞) (3x^2 + 2)/(5x^2 - 4x + 1)

Options:

Correct Answer: 3/5

Solution:

Dividing numerator and denominator by x^2 gives lim (x -> ∞) (3 + 2/x^2)/(5 - 4/x + 1/x^2) = 3/5.

Evaluate the limit: lim (x -> ∞) (3x^2 + 2)/(5x^2 - 4x + 1)

Evaluate the limit: lim (x -> ∞) (3x^2 + 2)/(5x^2 - 4x + 1)

Correct Answer: 3/5

Step 1: Identify the limit we want to evaluate: lim (x -> ∞) (3x^2 + 2)/(5x^2 - 4x + 1).
Step 2: Notice that both the numerator and the denominator are polynomials of degree 2 (the highest power of x is 2).
Step 3: To simplify the limit, 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 - 4x/x^2 + 1/x^2).
Step 5: Simplify each term: (3 + 2/x^2)/(5 - 4/x + 1/x^2).
Step 6: Now, evaluate the limit as x approaches infinity. As x becomes very large, the terms 2/x^2, -4/x, and 1/x^2 approach 0.
Step 7: This means the limit simplifies to (3 + 0)/(5 + 0) = 3/5.
Step 8: Therefore, the final answer is 3/5.

Limit Evaluation – Understanding how to evaluate limits as x approaches infinity, particularly for rational functions.
Dominant Terms – Identifying the dominant terms in the numerator and denominator when x is large.
Simplification Techniques – Using algebraic manipulation, such as dividing by the highest power of x, to simplify the limit.