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Chapter 2: Problem 13

If \(\lim _{x \rightarrow a^{-}} f(x)=L\) and \(\lim _{x \rightarrow a^{+}} f(x)=M,\) where \(L\) and \(M\) are finite real numbers, then how are \(L\) and \(M\) related if \(\lim f(x)\) exists?

Short Answer

Expert verified
Answer: If the limit of a function exists, the left-hand limit and the right-hand limit will be equal (L = M).

Step by step solution

01

Define Left-Hand and Right-Hand Limits

The left-hand limit of a function is the value that the function approaches as the input (x) approaches a certain value (a) from the left side (smaller values). Mathematically, it is represented as: $$\lim_{x \rightarrow a^{-}} f(x)=L$$ Similarly, the right-hand limit of a function is the value that the function approaches as the input (x) approaches the same value (a) from the right side (greater values). Mathematically, it is represented as: $$\lim_{x \rightarrow a^{+}} f(x)=M$$
02

Understand the Existence of a Limit

For a limit to exist at a point x=a, the left-hand limit and the right-hand limit must be equal to the same value. In other words, the function should approach the same value from both the left and the right sides as x approaches a. If this condition is met, then the limit of the function exists at x=a, and can be represented as: $$\lim_{x \rightarrow a} f(x)$$
03

Determine the Relationship Between L and M

Given the condition for the existence of a limit, if the limit of the function exists (as stated in the problem), then the left-hand limit (L) and the right-hand limit (M) must be equal. Therefore, the relationship between L and M can be stated as: $$L = M$$ So if the limit of a function exists, the left-hand limit and the right-hand limit will be equal (L = M).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Left-Hand Limit
The left-hand limit, often noted as \( \lim_{x \rightarrow a^{-}} f(x) \), is essential in understanding how a function behaves as it approaches a specific point from the left side. This means that as the input value \( x \) gets infinitesimally close to \( a \) but remains less than \( a \), the function approaches a certain value, which we call \( L \).
The concept of the left-hand limit helps us to comprehend the behavior of functions in scenarios where continuity is in question. If you imagine stepping closer to your destination from your left side, the left-hand limit represents your anticipated endpoint.
Key points to remember:
  • The function's behavior is examined from the side where \( x < a \).
  • The left-hand limit doesn't depend on what the function actually "does" at \( x=a \), only on what it approaches as \( x \) nears \( a \) from the left.
  • This concept is especially useful in situations where the function has a jump or discontinuity at \( x=a \).
Right-Hand Limit
The right-hand limit, symbolized as \( \lim_{x \rightarrow a^{+}} f(x) \), is the counterpart of the left-hand limit, but focuses on approaching a point from the right side. As \( x \) draws near to \( a \) but stays greater than \( a \), the function nears a specific value, which we denote as \( M \).
Visualize drawing closer to a destination from the right-hand side; the right-hand limit predicts where you'll end up.
Key aspects include:
  • Analyzes function behavior when \( x > a \).
  • Does not consider the exact value of the function at \( x=a \), just the trend as \( x \) gets close from the right.
  • Right-hand limits are crucial for functions with different behaviors on either side of \( a \).
Existence of Limit
A limit can only be said to exist at a point \( x=a \) if both the left-hand limit and the right-hand limit are equal and finite. Which means, \( L = M \).
This implies that as \( x \) approaches \( a \) from both sides, the function shouldn't just trend towards the same value, but it must consistently reach that exact value from both directions. This is a primary condition for continuity and smoothness in the behavior of functions.
Important considerations:
  • If the left-hand and right-hand limits equal each other, the function is well-behaved at that point, indicating potential continuity.
  • When a function has the same limit value coming from both left and right, it implies stability at \( x=a \).
  • If \( L eq M \), it indicates a discontinuity, such as a jump discontinuity, at \( x=a \).
  • Understanding the existence of limits helps determine if further analysis and solutions are viable at that point.

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Most popular questions from this chapter

Theorem 2.4 a Given the polynomial $$p(x)=b_{n} x^{n}+b_{n-1} x^{n-1}+\dots+b_{1} x+b_{0}$$ prove that \(\lim _{x \rightarrow a} p(x)=p(a)\) for any value of \(a\).

Find the following limits or state that they do not exist. Assume \(a, b, c,\) and k are fixed real numbers. $$\lim _{x \rightarrow 0} x \cos x$$

Find the following limits or state that they do not exist. Assume \(a, b, c,\) and k are fixed real numbers. $$\lim _{x \rightarrow a} \frac{x-a}{\sqrt{x}-\sqrt{a}}, a>0$$

Even function limits Suppose \(f\) is an even function where \(\lim _{x \rightarrow 1^{-}} f(x)=5\) and \(\lim _{x \rightarrow 1^{+}} f(x)=6 .\) Find \(\lim _{x \rightarrow-1^{-}} f(x)\) and \(\lim _{x \rightarrow-1^{+}} f(x)\).

Looking ahead to sequences A sequence is an infinite, ordered list of numbers that is often defined by a function. For example, the sequence \(\\{2,4,6,8, \ldots\\}\) is specified by the function \(f(n)=2 n,\) where \(n=1,2,3, \ldots . .\) The limit of such a sequence is \(\lim _{n \rightarrow \infty} f(n),\) provided the limit exists. All the limit laws for limits at infinity may be applied to limits of sequences. Find the limit of the following sequences or state that the limit does not exist. \(\left\\{0, \frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \ldots\right\\},\) which is defined by \(f(n)=\frac{n-1}{n},\) for \(n=1,2,3, \ldots\)
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