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

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

Step 1: Identify the limit we want to evaluate: lim(x->infinity) (2x^3 - 3x)/(4x^3 + 5).
Step 2: Notice that both the numerator and denominator have terms with x^3. To simplify, we will divide every term in the numerator and denominator by x^3.
Step 3: Rewrite the limit after dividing by x^3: lim(x->infinity) (2 - 3/x^2)/(4 + 5/x^3).
Step 4: As x approaches infinity, the terms 3/x^2 and 5/x^3 approach 0 because they become very small.
Step 5: Now the limit simplifies to: lim(x->infinity) (2 - 0)/(4 + 0) = 2/4.
Step 6: Finally, simplify 2/4 to get the result: 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.