Problem 90 - Brief Applied Calculus

Question

True or False: If \(\lim _{x \rightarrow 2} f(x)=7\), then \(\lim _{x \rightarrow 2^{+}} f(x)=7\)

Step-by-Step Solution

Answer

True

Step 1: Understanding the Problem

We are asked to determine if the statement is true or false. The statement claims that if the two-sided limit of a function \(f(x)\) as \(x\) approaches 2 is 7, then the right-hand limit as \(x\) approaches 2 from the positive side is also 7.

Step 2: Concept of Two-Sided Limits

The two-sided limit \(\lim_{x \to 2} f(x) = 7\) means that as \(x\) gets closer to 2 from both sides (left \(2^{-}\) and right \(2^{+}\)), the value of \(f(x)\) gets closer to 7.

Step 3: Implication of Two-Sided Limits

For a two-sided limit to exist and equal a certain value, both the left-hand and right-hand limits must exist and equal that value. Therefore, if \(\lim_{x \to 2} f(x) = 7\), then both \(\lim_{x \to 2^{+}} f(x)\) and \(\lim_{x \to 2^{-}} f(x)\) must equal 7.

Step 4: Conclusion

Since the condition \(\lim_{x \to 2} f(x) = 7\) implies that the right-hand limit also equals 7, the statement is true.

Key Concepts

Two-Sided Limits

Two-sided limits help us understand the behavior of a function as the input, denoted by \(x\), gets closer to a specific point from both directions, left and right. This is expressed mathematically as \(\lim_{x \to a} f(x) = L\), where \(L\) is the limit value when \(x\) approaches the point \(a\).

When we talk about a two-sided limit existing at a certain value, we mean that no matter how small we choose a number around \(a\), the function \(f(x)\) gets closer to \(L\) from both sides.

For example:

• If \(x\) approaches a value from the left (written as \(x \to a^{-}\)), and the function nears \(L\).
• And if \(x\) approaches from the right (written as \(x \to a^{+}\)), and the function also nears \(L\).

Only when both conditions are satisfied do we say the two-sided limit exists and equals \(L\). Hence, in our exercise, \(\lim_{x \to 2} f(x) = 7\) implies that from both left and right sides, the values of \(f(x)\) converge to 7.

Right-Hand Limits

The right-hand limit is specifically concerned with the behavior of a function \(f(x)\) as \(x\) approaches a particular point \(a\) from the positive, or right, side. It is symbolized as \(\lim_{x \to a^{+}} f(x)\).

This means that we are only interested in how \(f(x)\) behaves when \(x\) is slightly larger than \(a\). It does not consider values of \(x\) less than \(a\).

In the context of the exercise, we explore the right-hand limit \(\lim_{x \to 2^{+}} f(x)\). In this scenario, as \(x\) approaches 2 from values greater than 2, if \(f(x)\) gets closer to 7, the right-hand limit at \(a = 2\) is 7.

This aligns with the conclusion in the step-by-step solution: if the two-sided limit \(\lim_{x \to 2} f(x)\) is 7, the right-hand limit can also be deduced to be 7, confirming the statement as true.

Limit Existence

The existence of a limit is a crucial concept in calculus, determining whether a function approaches a particular value at a given point. When seeking to establish limit existence at a point \(a\), both the right-hand and left-hand limits must meet certain criteria:

• The right-hand limit, \(\lim_{x \to a^{+}} f(x)\), must exist.
• The left-hand limit, \(\lim_{x \to a^{-}} f(x)\), must also exist.


If both of these limits converge to the same value \(L\), we can say the two-sided limit exists, equaling \(L\), thus making the overall limit exist at that point.

In the exercise, the statement \(\lim_{x \to 2} f(x) = 7\) signifies complete limit existence because both one-sided limits (left and right) equate to the same value. This dual agreement confirms that the limit at \(x = 2\) indeed exists and affirms the answer to the question, making the given claim true.