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

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

Options:

Correct Answer: 2/5

Solution:

Divide numerator and denominator by x^2. The limit becomes lim (x -> ∞) (2 + 3/x^2)/(5 - 4/x + 1/x^2) = 2/5.

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

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

Step 1: Identify the limit you need to evaluate: lim (x -> ∞) (2x^2 + 3)/(5x^2 - 4x + 1).
Step 2: Notice that both the numerator and the denominator have terms with x^2. To simplify, divide every term in the numerator and the denominator by x^2.
Step 3: Rewrite the limit after dividing: (2x^2/x^2 + 3/x^2)/(5x^2/x^2 - 4x/x^2 + 1/x^2).
Step 4: Simplify each term: (2 + 3/x^2)/(5 - 4/x + 1/x^2).
Step 5: Now, evaluate the limit as x approaches infinity. As x becomes very large, 3/x^2 approaches 0, -4/x approaches 0, and 1/x^2 approaches 0.
Step 6: Substitute these values into the limit: (2 + 0)/(5 - 0 + 0) = 2/5.
Step 7: Conclude that the limit is 2/5.

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.