Problem 36 - Calculus of a Single Variable

Question

Finding a One-Sided Limit In Exercises \(33-48,\) find the one-sided limit (if it exists.). $$ \lim _{x \rightarrow 2^{-}} \frac{x^{2}}{x^{2}+4} $$

Step-by-Step Solution

Answer

The one-sided limit of the given function as x approaches 2 from the left is 0.5.

Step 1: Understand the function

The function is \( \frac{x^{2}}{x^{2}+4} \). We can write the limit as \( \lim _{x \rightarrow 2^{-}} \frac{x^{2}}{x^{2}+4} \), where the superscript '-' in \( x \rightarrow 2^{-} \) indicates that x is approaching 2 from the left.

Step 2: Substitute the limit point

Since there is no indeterminate form or discontinuity of the function at x=2, we can substitute x=2 directly into the function. This gives \( \frac{2^{2}}{2^{2}+4} = \frac{4}{4+4} \).

Step 3: Evaluate the function

Upon evaluating, we have \( \frac{4}{8} = 0.5 \). So, when x approaches 2 from the left, the function approaches 0.5.

Key Concepts

Limit Evaluation

Limit evaluation is considered one of the cornerstones in the study of calculus. It provides the foundation for understanding concepts like continuity, derivatives, and integrals. When we talk about evaluating a limit, we essentially want to describe the behavior of a function as the input value gets arbitrarily close to a certain point. Rather than focusing on the function's value at that point itself, limits allow us to answer what value the function is approaching.

Take, for example, the function \( \frac{x^{2}}{x^{2}+4} \). To evaluate the one-sided limit \( \lim _{x \rightarrow 2^{-}} \frac{x^{2}}{x^{2}+4} \), we closely observe the behavior of the function as the variable \(x\) approaches the number 2, but strictly from values less than 2. Unlike finding a general limit, we’re not concerned with what happens as x approaches 2 from values greater than 2. This distinction is crucial and changes how we interpret the function's behavior near the point of interest.

Approaching from the Left

Approaching a limit from the left is denoted by a superscript minus sign next to the point to which \(x\) is approaching—in our case, 2. This notation, \(x \rightarrow 2^{-}\), specifies that we are only interested in the values of \(x\) that are less than 2, that is on the left side of 2 on the real number line. When dealing with a one-sided limit from the left, one must consider the function's behavior exclusively from values less than the specified point.

Knowing this direction of approach is essential when dealing with functions that have different behaviors on either side of a point or for functions that aren't defined at the point. One-sided limits can reveal asymmetrical behavior around points of interest that full (two-sided) limits might obscure, thus providing deeper insight into the function's nature.

Indeterminate Form

In calculus, indeterminate forms occur when the limit of an expression is not immediately obvious, and direct substitution yields an uncertain or 'indeterminate' result. Classic examples of such forms include \(0/0\), \(\infty/\infty\), and \(\infty - \infty\). These forms require further manipulation or techniques such as L'Hopital's rule to evaluate.

Fortunately, not all limit evaluations involve indeterminate forms. In our exercise, substituting \(x=2\) into \(\frac{x^{2}}{x^{2}+4}\) does not yield an indeterminate form but a perfectly calculable number, \(\frac{4}{8}\) or \(0.5\). This means we can directly substitute the value of \(x\) to find the limit without more complex techniques. Identifying whether a limit expression results in an indeterminate form is a key step, and it simplifies the process if we do not encounter such forms.