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

₹0.0

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

Options:

Correct Answer: 3/5

Solution:

Dividing by x^2, lim(x→∞) (3 + 2/x^2)/(5 - 4/x^2) = 3/5.

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

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

Step 1: Identify the limit we want to evaluate: lim(x→∞) (3x^2 + 2)/(5x^2 - 4).
Step 2: Notice that both the numerator and the denominator 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: Rewrite the expression: (3x^2/x^2 + 2/x^2)/(5x^2/x^2 - 4/x^2).
Step 5: This simplifies to (3 + 2/x^2)/(5 - 4/x^2).
Step 6: Now, evaluate the limit as x approaches infinity: as x gets very large, 2/x^2 approaches 0 and 4/x^2 approaches 0.
Step 7: Substitute these values into the expression: (3 + 0)/(5 - 0).
Step 8: This gives us 3/5 as the final result.

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