Theory and Examples

a. If limx→0 f(x) / x² = 1, find limx→0 f(x).

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First, understand the given limit: \( \lim_{x \to 0} \frac{f(x)}{x^2} = 1 \). This means that as \( x \) approaches 0, the function \( \frac{f(x)}{x^2} \) approaches 1.

To find \( \lim_{x \to 0} f(x) \), consider the behavior of \( f(x) \) as \( x \) approaches 0. Since \( \frac{f(x)}{x^2} \to 1 \), it implies that \( f(x) \) behaves like \( x^2 \) near 0.

Multiply both sides of the equation \( \frac{f(x)}{x^2} = 1 \) by \( x^2 \) to isolate \( f(x) \): \( f(x) = x^2 \cdot 1 = x^2 \).

Now, substitute \( f(x) = x^2 \) into the limit expression: \( \lim_{x \to 0} f(x) = \lim_{x \to 0} x^2 \).

Evaluate the limit \( \lim_{x \to 0} x^2 \). As \( x \to 0 \), \( x^2 \to 0 \). Therefore, \( \lim_{x \to 0} f(x) = 0 \).

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Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. In this case, we are interested in the limit of f(x) as x approaches 0. Understanding limits is crucial for analyzing the continuity and behavior of functions at specific points.

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L'Hôpital's Rule is a method 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 derivative of the denominator. This rule can simplify the process of finding limits in complex scenarios.

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A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point. In this problem, if we find that limx→0 f(x) exists and equals a specific value, it indicates that f(x) is continuous at x = 0, which is essential for understanding the behavior of the function around that point.

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