Theory and Examples

If limx→−2 f(x) / x² = 1, find

b. limx→−2 f(x) / x

Verified step by step guidance

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

To find \( \lim_{x \to -2} \frac{f(x)}{x} \), consider the relationship between \( f(x) \) and \( x^2 \). Since \( \lim_{x \to -2} \frac{f(x)}{x^2} = 1 \), we can express \( f(x) \) as \( f(x) = x^2 + g(x) \), where \( g(x) \) is a function that becomes negligible as \( x \) approaches \( -2 \).

Substitute \( f(x) = x^2 + g(x) \) into \( \lim_{x \to -2} \frac{f(x)}{x} \) to get \( \lim_{x \to -2} \frac{x^2 + g(x)}{x} \).

Simplify the expression: \( \lim_{x \to -2} \left( x + \frac{g(x)}{x} \right) \). As \( x \to -2 \), \( \frac{g(x)}{x} \) should approach 0 if \( g(x) \) is negligible compared to \( x^2 \).

Finally, evaluate the limit: \( \lim_{x \to -2} x + \lim_{x \to -2} \frac{g(x)}{x} \). Since \( \lim_{x \to -2} \frac{g(x)}{x} \) approaches 0, the limit simplifies to \( \lim_{x \to -2} x \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 the function f(x) divided by x² as x approaches -2. Understanding limits is crucial for evaluating the behavior of functions near points of interest, especially when direct substitution may lead to indeterminate forms.

L'Hôpital's Rule

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, especially when dealing with rational functions.

Continuity

Continuity of a function at a point means that the function is defined at that point, the limit exists, and the limit equals the function's value at that point. In the context of the given problem, understanding whether f(x) is continuous at x = -2 will help in determining the limit of f(x)/x as x approaches -2. Continuity ensures that the behavior of the function is predictable near the point of interest.

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