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

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

Options:

Correct Answer: 1/2

Solution:

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

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

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

Step 1: Identify the limit we want to evaluate: lim (x -> ∞) (2x^3 - 3x)/(4x^3 + 5).
Step 2: Notice that both the numerator and denominator have terms with x^3, which is the highest power of x in both.
Step 3: To simplify, divide every term in the numerator and the denominator by x^3.
Step 4: Rewrite the limit: lim (x -> ∞) (2 - 3/x^2)/(4 + 5/x^3).
Step 5: As x approaches infinity, the terms 3/x^2 and 5/x^3 approach 0.
Step 6: Substitute these values into the limit: lim (x -> ∞) (2 - 0)/(4 + 0) = 2/4.
Step 7: Simplify 2/4 to get the final answer: 1/2.

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 approaches infinity.
Simplification Techniques – Using algebraic manipulation, such as dividing by the highest power of x, to simplify the limit expression.