Question

Prove the following statements to establish the fact that \(\lim _{x \rightarrow a} f(x)=L\) if and only if \(\lim _{x \rightarrow a^{-}} f(x)=L\) and \(\lim _{x \rightarrow a^{+}} f(x)=L\) a. If \(\lim _{x \rightarrow a^{-}} f(x)=L\) and \(\lim _{x \rightarrow a^{+}} f(x)=L,\) then \(\lim _{x \rightarrow a} f(x)=L\) b. If \(\lim _{x \rightarrow a} f(x)=L,\) then \(\lim _{x \rightarrow a^{-}} f(x)=L\) and \(\lim _{x \rightarrow a^{+}} f(x)=L\)

Short answer

Question: Prove that the limit of a function \(f(x)\) as \(x\) approaches \(a\) is \(L\) if and only if the left-hand limit and the right-hand limit at the point \(a\) are also equal to \(L\). Answer: This statement is true as we have proven in the solution above that (a) if the left-hand limit and right-hand limit are equal to L, then the limit is also L, and (b) if the limit is L, then the left-hand limit and right-hand limit are also equal to L.

Step-by-step solution

Part a: Prove that if the left-hand limit and right-hand limit are equal to L, then the limit is also L

To prove this, let's assume that \(\lim_{x \rightarrow a^-} f(x)=L\) and \(\lim_{x \rightarrow a^+} f(x)=L\). We want to show that \(\lim_{x \rightarrow a} f(x)=L\). We use the definition of the left-hand limit and the right-hand limit, which states that for any given \(\epsilon >0\), there exist \(\delta_1>0\) and \(\delta_2>0\) such that if \(0<|x-a|<\delta_1\) and \(xa\), then \(|f(x)-L|<\epsilon\). Now, let's define \(\delta = \min\{\delta_1, \delta_2\}\). Therefore, for \(0<|x-a|<\delta\), we have: 1. If \(xa\), then \(|f(x)-L|<\epsilon\) because \(|x-a|<\delta_2\) Thus, combining both cases, we can conclude that \(\lim_{x \rightarrow a} f(x)=L\).

Part b: Prove that if the limit is L, then the left-hand limit and right-hand limit are also equal to L

To prove this part, let's assume that \(\lim_{x \rightarrow a} f(x)=L\). We want to show that \(\lim_{x \rightarrow a^-} f(x)=L\) and \(\lim_{x \rightarrow a^+} f(x)=L\). First, we show that \(\lim_{x \rightarrow a^-} f(x)=L\). Using the definition of the limit, it means that for any given \(\epsilon>0\), there exists a \(\delta>0\) such that if \(0<|x-a|<\delta\), then \(|f(x)-L|<\epsilon\). Now, let's consider \(xa\). Since \(0<|x-a|<\delta\), we have \(|f(x)-L|<\epsilon\). Thus, we can conclude that \(\lim_{x \rightarrow a^+} f(x)=L\). In conclusion, we have proven that \(\lim_{x \rightarrow a} f(x)=L\) if and only if \(\lim_{x \rightarrow a^-} f(x)=L\) and \(\lim_{x \rightarrow a^+} f(x)=L\).

Key concepts

Continuity

Imagine a road. If you can walk along it without ever having to jump or stop, it's continuous. Similarly, a function is continuous at a point if you can draw it without lifting your pencil. Mathematically, this means that at the point \(a\), the limit of the function as \(x\) approaches \(a\) must equal the function value at \(a\).
In symbols, a function \(f(x)\) is continuous at a point \(a\) if:

• \(\lim_{x \rightarrow a} f(x) = f(a)\)

Continuity ensures there are no breaks, jumps, or holes in the function at that point. For students, grasping continuity helps in understanding concepts like the Intermediate Value Theorem, which relies on the uninterrupted nature of functions. Remember, just because a function looks smooth, doesn't mean it aligns perfectly at every point. Always check the mathematical definitions.

One-Sided Limits

In calculus, limits are like peeking around the corner. One-sided limits focus on viewing from one direction at a time. If a function approaches the same value from both the left and the right as \(x\) approaches a point \(a\), then the overall limit exists.
There are two types of one-sided limits:

• Left-Hand Limit: Notated as \(\lim_{x \rightarrow a^-} f(x)\), it represents the value the function approaches from numbers less than \(a\).
• Right-Hand Limit: Notated as \(\lim_{x \rightarrow a^+} f(x)\), it captures the function's behavior approaching from numbers greater than \(a\).

When these one-sided limits match at \(a\), we say that the overall limit, \(\lim_{x \rightarrow a} f(x)\), also exists and equals the matched value. If either limit behaves differently, the function has a jump discontinuity, often signaling a break or an abrupt change at that point.

Epsilon-Delta Definition

The epsilon-delta definition is like the secret code of calculus. It's a formal way to define what it means for a limit to exist. The basic idea is about being as close as you want to a specific value — within a positive distance, "epsilon," of the limit \(L\).
To establish that \(\lim_{x \rightarrow a} f(x) = L\), we need to show:

• For every \(\epsilon > 0\), there's a corresponding \(\delta > 0\)
• Such that if \(0 < |x - a| < \delta\), then \(|f(x) - L| < \epsilon\)

This allows us to handle limits rigorously. Whenever we're working with \(\epsilon\) and \(\delta\), we're essentially proving that no matter how sluggish or strict our precision for \(f(x)\) is, we can always match it by pinpointing \(x\) close enough to \(a\). This provides the foundation for many proofs in calculus, ensuring everyone is speaking the same 'mathematical language.'