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

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

Options:

Correct Answer: 3/5

Solution:

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

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

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

Step 1: Identify the limit we want to find: lim (x -> ∞) (3x^2 + 2)/(5x^2 - 4).
Step 2: Notice that both the numerator (3x^2 + 2) and the denominator (5x^2 - 4) have the highest power of x as x^2.
Step 3: To simplify, divide every term in the numerator and the denominator by x^2.
Step 4: After dividing, the numerator becomes (3 + 2/x^2) and the denominator becomes (5 - 4/x^2).
Step 5: Now we rewrite the limit: lim (x -> ∞) (3 + 2/x^2)/(5 - 4/x^2).
Step 6: As x approaches infinity, the terms 2/x^2 and 4/x^2 approach 0.
Step 7: Therefore, the limit simplifies to (3 + 0)/(5 - 0) = 3/5.
Step 8: Conclude that the limit is 3/5.

Limit at Infinity – 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 very large.
Simplifying Rational Functions – Using algebraic manipulation, such as dividing by the highest power of x, to simplify the limit calculation.