Question

Limits Involving Zero or Infinity $$\lim _{x \rightarrow 0}\left(\frac{1}{2+x}-\frac{1}{2}\right) \cdot \frac{1}{x}$$

Short answer

The limit as x approaches 0 of the expression is -1/4.

Step-by-step solution

Factor the expression within the limit

Combine the fractions in the numerator by finding a common denominator. Do not alter the denominator.

Simplify the expression

Simplify the expression by combining the numerators over the common denominator, then simplify the entire fraction by dividing the numerator by the common denominator.

Apply the limit

Evaluate the limit by substituting the value of x approaching zero into the simplified expression, if possible.

Simplify the result

If the substitution in the previous step leads to a determinate form, simplify to get the final result. If the expression is in an indeterminate form, further steps such as L'Hopital's Rule may be necessary.

Key concepts

Limit Evaluation

Understanding how to evaluate limits, especially when they involve zero or infinity, is a fundamental skill in calculus. To evaluate a limit means to find the value that a function approaches as the input approaches a certain value. For instance, the given limit \[ \lim _{x \rightarrow 0}\left(\frac{1}{2+x}-\frac{1}{2}\right) \cdot \frac{1}{x} \] asks what value the expression approaches as 'x' approaches zero. Sometimes, this is straightforward and can be done by direct substitution. However, when direct substitution leads to an undefined expression or an indeterminate form such as 0/0 or ∞/∞, alternative methods must be used to find the limit.

It's essential to simplify expressions where possible before applying the limit, which we see in the steps of the given solution. This process typically involves factoring, expanding, or combining terms to get a form that makes limit evaluation easier.

Simplifying Expressions

Simplifying expressions is crucial before attempting to evaluate limits that involve complex fractions or algebraic expressions. Simplification can involve finding common denominators, reducing fractions, or expanding and factoring polynomial expressions.

The goal is to transform the original complex expression into a simpler form that allows for direct limit evaluation or the application of other techniques such as L'Hopital's Rule. In the provided exercise, we consolidate the fractions by finding a common denominator, which prevents undefined conditions and reveals the behavior of the function as 'x' approaches zero. Without simplification, it would be challenging to see how the function behaves near the problematic point and to apply further calculus tools effectively.

L'Hopital's Rule

L'Hopital's Rule is a rescue tool in calculus used to resolve indeterminate expressions like 0/0 or ∞/∞ during limit evaluations. It states that under certain conditions, if the limit of a function \[ \lim _{x \rightarrow a} \frac{f(x)}{g(x)} \] results in an indeterminate form, then it can be evaluated as \[ \lim _{x \rightarrow a} \frac{f'(x)}{g'(x)} \], where f'(x) and g'(x) are the derivatives of f(x) and g(x) respectively.

This rule can only be applied when both f(x) and g(x) are differentiable near 'a' and g'(x) does not equal zero at 'a'. In practice, L'Hopital's Rule often involves taking derivatives one or more times until a determinate form emerges, allowing for the limit to be computed.

Indeterminate Forms

Indeterminate forms are expressions in calculus that do not have a clearly defined value. Common examples include 0/0, ∞/∞, ∞ - ∞, 0⋅∞, 1^∞, ∞^0, and 0^0. These forms typically arise during limit evaluations and pose a unique challenge because they don't inherently reveal the limit's value.

In our exercise, the process of simplification and subsequent evaluation could lead to an indeterminate form, which signals that further analysis is necessary to find the limit. Techniques such as factoring, expanding, using conjugates, or applying L'Hopital's Rule, as mentioned, are essential for resolving indeterminate forms and determining the precise behavior of a function as it approaches a particular point. Recognizing and dealing with indeterminate forms are key to mastering limit evaluations in calculus.