Problem 5 - Calculus Early Transcendentals

Question

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?

Step-by-Step Solution

Answer

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

Step 1: 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: \[lim_{x \to a^{-}} f(x) = lim_{x \to a^{+}} f(x) = L\]

Step 2: 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: \[lim_{x \to a^{+}} f(x) = M\]

Step 3: 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: \[L = M\]

Step 4: 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: \[L = M\]

Key Concepts

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.