Question

evaluate the limit \\(\lim_{x \to \infty} \frac{10x^2 - 11x + 4}{7x + 10}\\)

Explanation:

Step1: Analyze Degree of Polynomials

The numerator \(10x^2 - 11x + 4\) is a quadratic polynomial (degree 2), and the denominator \(7x + 10\) is a linear polynomial (degree 1). For limits as \(x \to \infty\) of a rational function \(\frac{f(x)}{g(x)}\), if \(\text{deg}(f(x)) > \text{deg}(g(x))\), the limit is \(\pm\infty\) (sign depends on leading coefficients).

Step2: Check Leading Coefficients

Leading term of numerator: \(10x^2\) (leading coefficient \(10>0\)), leading term of denominator: \(7x\) (leading coefficient \(7>0\)). Since degree of numerator (2) > degree of denominator (1) and leading coefficients are positive, the limit will be \(+\infty\).

Answer:

\(\infty\) (or "infinity")