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Evaluate the limit: lim (x -> ∞) (3x^2 + 2)/(5x^2 - 4x + 1)
Evaluate the limit: lim (x -> ∞) (3x^2 + 2)/(5x^2 - 4x + 1)
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Practice Questions
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Q1
Evaluate the limit: lim (x -> ∞) (3x^2 + 2)/(5x^2 - 4x + 1)
3/5
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Dividing numerator and denominator by x^2 gives lim (x -> ∞) (3 + 2/x^2)/(5 - 4/x + 1/x^2) = 3/5.
Questions & Step-by-step Solutions
1 item
Q
Q: Evaluate the limit: lim (x -> ∞) (3x^2 + 2)/(5x^2 - 4x + 1)
Solution:
Dividing numerator and denominator by x^2 gives lim (x -> ∞) (3 + 2/x^2)/(5 - 4/x + 1/x^2) = 3/5.
Steps: 8
Show Steps
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.
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