Problem 105 - Calculus: Early Transcendentals

Question

Suppose \(g(x)=f(1-x)\) for all \(x, \lim _{x \rightarrow 1^{+}} f(x)=4,\) and \(\lim _{x \rightarrow 1^{-}} f(x)=6 .\) Find \(\lim _{x \rightarrow 0^{+}} g(x)\) and \(\lim _{x \rightarrow 0^{-}} g(x)\).

Step-by-Step Solution

Answer

Answer: The right-hand limit of g(x) as x approaches 0 is 6, and the left-hand limit of g(x) as x approaches 0 is 4.

Step 1: Define g(x) and Find Its Limits

We are given that \(g(x) = f(1 - x)\). We want to find \(\lim_{x \rightarrow 0^{+}} g(x)\) and \(\lim_{x \rightarrow 0^{-}} g(x)\).

Step 2: Substitute x with 1 - x

Let's make the substitution \(u = 1 - x\). Since we want to find the right-hand limit of g(x), we need to see how u behaves when x approaches 0 from the right, or in other words when \(x \rightarrow 0^{+}\). Under this condition, \(u = 1 - x\) will approach 1 from the left, which is denoted as \(u \rightarrow 1^{-}\). Similarly, if x approaches 0 from the left, or \(x \rightarrow 0^{-}\), then \(u = 1 - x\) will approach 1 from the right, denoted as \(u \rightarrow 1^{+}\).

Step 3: Apply the Limits to g(x)

Now, we can apply the limits to g(x) in terms of u: $$\lim_{x \rightarrow 0^{+}} g(x) = \lim_{u \rightarrow 1^{-}} f(u) = 6$$ And $$\lim_{x \rightarrow 0^{-}} g(x) = \lim_{u \rightarrow 1^{+}} f(u) = 4$$ So we have our final answers:

Step 4: Find the Right-Hand and Left-Hand Limits

The right-hand limit of g(x) as x approaches 0 is 6: $$\lim_{x \rightarrow 0^{+}} g(x) = 6$$ And the left-hand limit of g(x) as x approaches 0 is 4: $$\lim_{x \rightarrow 0^{-}} g(x) = 4$$

Key Concepts

Limit from the Right

When we talk about finding the "Limit from the Right," we mean analyzing the behavior of a function as the input value approaches a specific number from values greater than that number. In mathematical terms, for a function \(f(x)\), if we are considering the limit as \(x\) approaches a number \(c\) from the right, we write it as \(\lim_{x \rightarrow c^{+}} f(x)\). Here, \(x\) gets closer to \(c\) by taking values that are slightly larger than \(c\).

In the original exercise, we have a function \(g(x) = f(1 - x)\) and we need to find \(\lim_{x \rightarrow 0^{+}} g(x)\). To do this, we substitute \(u = 1 - x\). As \(x\) approaches \(0\) from the right, \(u\) approaches \(1\) from the left. This means we are evaluating \(\lim_{u \rightarrow 1^{-}} f(u)\).

According to the given information, \(\lim_{x \rightarrow 1^{-}} f(x) = 6\), which means those approaching from values less than 1 lead \(f(x)\) to 6. Therefore, \(\lim_{x \rightarrow 0^{+}} g(x) = 6\).

• Identify the input approaching from the right.
• Make substitution adjustments if necessary.
• Apply the known limits of the original function \(f(x)\).

Limit from the Left

The "Limit from the Left" involves understanding how a function behaves as its input approaches a specific number from smaller values. For a function \(f(x)\), the limit as \(x\) approaches \(c\) from the left is written as \(\lim_{x \rightarrow c^{-}} f(x)\). Here, \(x\) nears \(c\) from values that are slightly less than \(c\).

In the problem, we determined \(g(x)\) as \(f(1 - x)\) and sought \(\lim_{x \rightarrow 0^{-}} g(x)\). By setting \(u = 1 - x\), as \(x\) nears \(0\) from the left, \(u\) nears \(1\) from the right, transitioning us to \(\lim_{u \rightarrow 1^{+}} f(u)\).

From the problem's premise, \(\lim_{x \rightarrow 1^{+}} f(x) = 4\), indicating that as we approach 1 from greater values, \(f(x)\) equals 4. Thus, \(\lim_{x \rightarrow 0^{-}} g(x) = 4\).

• Identify approaching values from the left side.
• Make basic substitutions possibly required.
• Utilize provided limits for \(f(x)\) to solve \(g(x)\).

Substitution Method

The substitution method can simplify complex limit problems by transforming a function into a more manageable form. This tactic involves substituting a part of the expression with a different variable, making it easier to analyze the limit.

In the given task, \(g(x) = f(1 - x)\) required finding its limit as \(x\) approached both sides of zero. The substitution \(u = 1 - x\) was crucial. It allowed converting the problem from finding the limit of \(g(x)\) as \(x\) neared 0 to a more straightforward limit involving \(f(u)\) as \(u\) approached 1, either from the left or right.

Steps to employ substitution effectively include:

• Identify parts of the function that complicate taking a limit, such as "1 - x" in this exercise.
• Introduce a simpler variable such as \(u = 1 - x\). This aligns the problem with the known behaviors of \(f(x)\).
• Re-evaluate the limit in terms of new variables to use existing information or limits.

Substitution often turns challenging limit problems into simpler ones through strategic variable alteration, as seen here with transition adjustments and limit applications.