Consider the lim_x→∞ (√ ax + b) / √cx + d where a, b, c, and d are positive real numbers. Show that l’Hôpital’s Rule fails for this limit. Find the limit using another method.

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Identify the form of the limit: As x approaches infinity, both the numerator and the denominator approach infinity, creating an indeterminate form of type ∞/∞.

Apply l'Hôpital's Rule: Differentiate the numerator and the denominator with respect to x. The derivative of the numerator √(ax + b) is (a / (2√(ax + b))), and the derivative of the denominator √(cx + d) is (c / (2√(cx + d))).

Evaluate the new limit: The limit of the derivatives as x approaches infinity is (a / (2√(ax + b))) / (c / (2√(cx + d))). Simplify this expression to (a/c) * (√(cx + d) / √(ax + b)).

Observe the behavior of the simplified expression: As x approaches infinity, the expression (√(cx + d) / √(ax + b)) approaches √(c/a) because the terms b and d become negligible compared to ax and cx.

Conclude that l'Hôpital's Rule fails: The application of l'Hôpital's Rule does not resolve the indeterminate form, as the limit of the derivatives still results in an indeterminate form. Instead, use the dominant term method to find the limit, which is √(a/c).

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L'Hôpital's Rule

L'Hôpital's Rule is a method in calculus used to evaluate limits of indeterminate forms, such as 0/0 or ∞/∞. It states that if the limit of f(x)/g(x) results in an indeterminate form, the limit can be found by taking the derivative of the numerator and the denominator separately. However, this rule may not apply if the derivatives do not yield a determinate form or if the limit diverges.

Limits at Infinity

Limits at infinity involve evaluating the behavior of a function as the input approaches infinity. In this context, we analyze how the function behaves as x becomes very large. Understanding how to simplify expressions by focusing on the highest degree terms in polynomials or radical expressions is crucial for finding these limits.

Dominant Terms

In the context of limits, dominant terms refer to the terms in a function that have the greatest influence on its behavior as x approaches a certain value, such as infinity. For rational functions or expressions involving radicals, identifying these terms allows for simplification, making it easier to evaluate the limit. This concept is essential for determining the limit without relying on L'Hôpital's Rule.