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Chapter 2: Problem 5

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 \(\lim _{x \rightarrow a} f(x)\) exists?

Short Answer

Expert verified
Answer: When the limit exists, the left limit (L) and the right limit (M) must be equal, i.e., L = M.

Step by step solution

01

Definition of Existence of a Limit

In order for the limit of a function as x approaches a to exist, the function must approach the same value L from both the left (a-) and the right (a+). Mathematically, this can be expressed as: \[lim_{x \to a^{-}} f(x) = lim_{x \to a^{+}} f(x) = L\]
02

Relationship between Left and Right Limits

Since we are given that the limit of the function as x approaches a exists and is equal to L, we can write: \[lim_{x \to a} f(x) = L\] We are also given that the limit of the function as x approaches a from the right (a+) exists and is equal to M: \[lim_{x \to a^{+}} f(x) = M\]
03

Equating Both Sides

Since the limit exists and is equal to L, the function approaches the same value L from both the left and the right. Therefore, we can equate the given expressions, which results in: \[lim_{x \to a^{-}} f(x) = lim_{x \to a^{+}} f(x)\] This implies that: \[L = M\]
04

Relationship between L and M

In conclusion, when the limit exists for a given function f(x) as x approaches a, the limit from the left (L) and the limit from the right (M) must 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
When we discuss the left-hand limit of a function, we consider the behavior of the function as it gets closer to a target point from the left, or the negative side. The notation used is \( \lim_{x \to a^{-}} f(x) \). This represents the value that the function \( f(x) \) approaches as the input \( x \) comes closer to a specific point \( a \) from values less than \( a \).

For the left-hand limit to exist, \( f(x) \) must converge to a single, finite value as \( x \) gets nearer to \( a \) from the left.
  • Example: Consider the function \( f(x) = 2 \) for all \( x < a \). The left-hand limit of this function as \( x \to a^{-} \) is 2, as it's evident the function remains constant as \( x \) approaches \( a \) from the left.
Understanding the left-hand limit is crucial in evaluating if the limit at point \( a \) exists.
Right-Hand Limit
The right-hand limit focuses on observing how a function behaves as it approaches a specific point from the right, or positive side. We denote this using \( \lim_{x \to a^{+}} f(x) \). This limit indicates the value the function \( f(x) \) gets closer to as \( x \) approaches \( a \) from values greater than \( a \).

For a right-hand limit to be well-defined, the function should approach the same number from the right side without oscillating or diverging.
  • Example: If \( f(x) = 3 \) for \( x > a \), then the right-hand limit of \( f(x) \) as \( x \to a^{+} \) is 3, because it stabilizes at that value.
Just like the left-hand limit, the right-hand limit is integral in determining the overall limit at any given point.
Continuity
Continuity at a point guarantees a smooth transition of function values through that point. For a function \( f(x) \) to be continuous at \( x = a \), three conditions must be met:
  • The function \( f(x) \) must be defined at \( x = a \), meaning \( f(a) \) must exist.
  • \( \lim_{x \to a^{-}} f(x) = \lim_{x \to a^{+}} f(x) \), meaning the left-hand limit equals the right-hand limit.
  • The limit \( \lim_{x \to a} f(x) \) must equal \( f(a) \).
These criteria ensure there is no sudden jump or discontinuity at the point \( a \). If these conditions hold, \( f(x) \) is considered continuous at \( a \), and there are no breaks, jumps, or holes in its behavior as \( x \) approaches \( a \). Understanding continuity is essential for comprehending not just limits at a point, but also the overall behavior of functions across their domains.

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Most popular questions from this chapter

Evaluate the following limits, where \(c\) and \(k\) are constants. \(\lim _{x \rightarrow 2}(5 x-6)^{3 / 2}\)

Suppose \(f\) is defined for all values of \(x\) near \(a\) except possibly at \(a .\) Assume for any integer \(N>0\) there is another integer \(M>0\) such that \(|f(x)-L|<1 / N\) whenever \(|x-a|<1 / M .\) Prove that \(\lim _{x \rightarrow a} f(x)=L\) using the precise definition of a limit.

Evaluate the following limits or state that they do not exist. $$\lim _{x \rightarrow 0^{+}} \frac{x}{\ln x}$$

Determine the value of the constant \(a\) for which the function $$f(x)=\left\\{\begin{array}{ll} \frac{x^{2}+3 x+2}{x+1} & \text { if } x \neq-1 \\\a & \text { if } x=-1\end{array}\right.$$ is continuous at -1.

Suppose \(g(x)=\left\\{\begin{array}{ll}x^{2}-5 x & \text { if } x \leq-1 \\ a x^{3}-7 & \text { if } x>-1.\end{array}\right.\) Determine a value of the constant \(a\) for which \(\lim _{x \rightarrow-1} g(x)\) exists and state the value of the limit, if possible.
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