Evaluating a Limit as x Approaches Infinity

This question focuses on evaluating a limit at infinity where two large, similar terms are being subtracted: a square root of a quadratic expression in x and a linear term. The candidate must recognize that, although each piece grows without bound, their difference can remain finite, and that the raw expression looks like an indeterminate form. It sits in the standard family of calculus questions that probe intuition about growth rates and asymptotics, often seen in early screening for roles that expect fluent manipulation of algebraic expressions and limits.

To solve it cleanly, the candidate needs to use conjugate multiplication to rationalize the difference of square roots, turning it into a quotient where cancellation exposes the effective leading-order behavior. The problem leans on understanding dominant terms in polynomials, square root expansions, and careful limit evaluation as x becomes large. Interviewers watch for algebraic precision, the ability to choose an appropriate manipulation without being prompted, and a clear explanation of why the apparent indeterminate form actually hides a simple finite limit.

What it tests

When evaluating limits involving expressions like $\sqrt{x^2 + bx} - x$ as $x \to \infty$, the key structure is the competition between terms of similar leading order in $x$. The apparent indeterminate form arises because both terms grow large, but their difference can converge to a finite value due to cancellation of the dominant $x^2$ terms. Rationalizing the expression (multiplying by the conjugate) exposes the true asymptotic behavior by converting the difference of roots into a difference of squares, which simplifies the leading order terms and reveals the next significant term. This pattern holds because, for large $x$, $\sqrt{x^2 + bx}$ behaves like $x + b/2$ plus lower-order terms, so subtracting $x$ isolates the finite offset. The principle is that subtracting two nearly equal, large quantities often requires algebraic manipulation to see the finite limiting behavior.