Question

Suppose that \(f\) is bounded on an open interval containing \(x_{0}\). Show that \(\lim _{x \rightarrow x_{0}} f(x)\) exists if and only if $$ \lim _{x \rightarrow x_{0}-} f(x)=\lim _{x \rightarrow x_{0}+} f(x)=\varliminf_{x \rightarrow x_{0}-} f(x)=\lim _{x \rightarrow x_{0}+} f(x) . $$ in which case \(\lim _{x \rightarrow \infty 0} f(x)\) is the common value of these four expressions.

Short answer

Question: Show that the limit of a function exists at a given point if and only if the left and right limits exist and are equal. Answer: To prove this, we've demonstrated that if the limit exists, then the left and right limits must exist and be equal. Conversely, if the left and right limits exist and are equal, then the overall limit exists. Therefore, the limit of a function exists at a given point if and only if the left and right limits exist and are equal.

Step-by-step solution

Part 1: If \(\lim_{x \rightarrow x_0} f(x)\) exists, then \(\lim_{x \rightarrow x_0^-} f(x)\) and \(\lim_{x \rightarrow x_0^+} f(x)\) exist and are equal.

We are given that \(\lim_{x \rightarrow x_0} f(x) = L\). By definition, for every \(\epsilon > 0\), there exists a \(\delta > 0\) such that if \(0 < |x - x_0| < \delta\), then \(|f(x) - L| < \epsilon\). Let's consider the left and right limits separately. For the left limit, we look at values of \(x\) that are close to \(x_0\) but are to the left of it (i.e., with \(x < x_0\)). If \(0 < x_0 - x < \delta\), then \(|x - x_0| < \delta\). By the definition of the limit, we have \(|f(x) - L| < \epsilon\). Thus, we've shown that for every \(\epsilon > 0\), there exists a \(\delta > 0\) such that if \(0 < x_0 - x < \delta\), then \(|f(x) - L| < \epsilon\). Therefore, \(\lim_{x \rightarrow x_0^-} f(x) = L\). For the right limit, we look at values of \(x\) that are close to \(x_0\) but are to the right of it (i.e., with \(x > x_0\)). If \(0 < x - x_0 < \delta\), then \(|x - x_0| < \delta\). By the definition of the limit, we have \(|f(x) - L| < \epsilon\). Thus, we've shown that for every \(\epsilon > 0\), there exists a \(\delta > 0\) such that if \(0 < x - x_0 < \delta\), then \(|f(x) - L| < \epsilon\). Therefore, \(\lim_{x \rightarrow x_0^+} f(x) = L\). Since both left and right limits exist and are equal to \(L\), then \(\lim_{x \rightarrow x_0^-} f(x) = \lim_{x \rightarrow x_0^+} f(x) = L\).

Part 2: If \(\lim_{x \rightarrow x_0^-} f(x)\) and \(\lim_{x \rightarrow x_0^+} f(x)\) exist and are equal, then \(\lim_{x \rightarrow x_0} f(x)\) exists.

We are given that \(\lim_{x \rightarrow x_0^-} f(x) = \lim_{x \rightarrow x_0^+} f(x) = L\). By definition, for every \(\epsilon > 0\), there exist \(\delta_1, \delta_2 > 0\) such that: 1. If \(0 < x_0 - x < \delta_1\), then \(|f(x) - L| < \epsilon\). 2. If \(0 < x - x_0 < \delta_2\), then \(|f(x) - L| < \epsilon\). Now, let \(\delta = \min\{\delta_1, \delta_2\}\). If \(0 < |x - x_0| < \delta\), then either \(0 < x_0 - x < \delta_1\) or \(0 < x - x_0 < \delta_2\). In either case, we have \(|f(x) - L| < \epsilon\). Thus, we've shown that for every \(\epsilon > 0\), there exists a \(\delta > 0\) such that if \(0 < |x - x_0| < \delta\), then \(|f(x) - L| < \epsilon\). Therefore, \(\lim_{x \rightarrow x_0} f(x)\) exists and is equal to \(L\).

Key concepts

Understanding Limits

In real analysis, the concept of limits is fundamental. Limits help us understand the behavior of a function as the input approaches a particular point. Simply put, if we say \(\lim_{x \rightarrow x_0} f(x) = L\), it means that as \(x\) gets closer to \(x_0\), the function \(f(x)\) gets closer to the value \(L\).

It is crucial to remember that a limit describes the trend of the function, not necessarily the actual value at that point. Therefore, it’s possible for a function to have a limit at a point even if it is not defined exactly at that point.

The process of finding limits involves understanding both the left-hand and right-hand limits:

• Left-hand limit \((\lim_{x \rightarrow x_0^-} f(x))\): Examines values of \(x\) approaching \(x_0\) from the left.
• Right-hand limit \((\lim_{x \rightarrow x_0^+} f(x))\): Considers values of \(x\) as they approach \(x_0\) from the right.

For a limit \(\lim_{x \rightarrow x_0} f(x)\) to exist, both the left-hand and right-hand limits must exist and be equal. This highlights the importance of checking both directions when evaluating limits.

Epsilon-Delta Definition

The epsilon-delta definition is a formal way to define the limit of a function in calculus. It captures the essence of what it means for a function to approach a particular value.

Here's a step-by-step breakdown:

• We start by considering the statement \(\lim_{x \rightarrow x_0} f(x) = L\).
• For every small positive number \(\epsilon\) (epsilon represents how close you're willing to be to \(L\)), there exists a small positive number \(\delta\) such that if \(0 < |x - x_0| < \delta\), then \(|f(x) - L| < \epsilon\).
• This definition implies that we can make \(f(x)\) as close as we want to \(L\) (within \(\epsilon\)) by choosing \(x\) sufficiently close to \(x_0\) (within \(\delta\)).

Essentially, the epsilon-delta definition is about controlling the outputs of a function by restricting its inputs.

Understanding Bounded Functions

A function is said to be bounded on an interval if there is a limit to its values within that interval — it cannot go to infinity or negative infinity within that space. For any bounded function, you can find a real number such that the function's output never exceeds this number.

Boundedness is an important property because it prevents the function from having excessive growth in value. It makes proving or refuting the existence of limits more feasible, as bounded functions are more predictable across an interval.

In the context of the exercise, knowing that \(f\) is bounded ensures that examining its behavior around \(x_0\) will not be thwarted by unbounded growth or decrease, which helps us apply our limit and epsilon-delta principles effectively.