Determine the following limits.
lim x→∞ (2x − 3) / (4x + 10)

Verified step by step guidance

1. Identify the highest degree terms in the numerator and the denominator. In this case, both are linear terms: \$2x\( in the numerator and \)4x$ in the denominator.
2. Divide every term in the numerator and the denominator by \(x\), the highest power of \(x\) present in the expression.
3. Rewrite the expression as \(\frac{2x/x - 3/x}{4x/x + 10/x}\), which simplifies to \(\frac{2 - 3/x}{4 + 10/x}\).
4. As \(x\) approaches infinity, the terms \(3/x\) and \(10/x\) approach zero.
5. Evaluate the limit of the simplified expression \(\frac{2 - 0}{4 + 0}\), which simplifies to \(\frac{2}{4}\).

Key Concepts

Limits at Infinity

Limits at infinity involve evaluating the behavior of a function as the input approaches infinity. This concept is crucial for understanding how functions behave for very large values of x, which can help determine horizontal asymptotes and overall end behavior.

Rational Functions

A rational function is a ratio of two polynomials. In the limit problem presented, recognizing that both the numerator and denominator are polynomials allows us to simplify the expression by focusing on the leading terms, which dominate the behavior as x approaches infinity.

Leading Coefficients

The leading coefficients of the highest degree terms in the numerator and denominator play a key role in determining the limit of a rational function as x approaches infinity. For the limit in question, the leading terms (2x and 4x) dictate the limit's value, allowing for straightforward simplification.