Problem 53 - A Graphical Approach to Precalculus with Limits

Question

Evaluate each limit. (a) \(\lim _{x \rightarrow 1^{-}} \sqrt{1-x}\) (b) \(\lim _{x \rightarrow 1^{+}} \sqrt{1-x}\) (c) \(\lim _{x \rightarrow 1} \sqrt{1-x}\)

Step-by-Step Solution

Answer

(a) 0; (b) does not exist; (c) does not exist.

Step 1: Understand One-Sided Limit A

For the limit \( \lim_{x \rightarrow 1^{-}} \sqrt{1-x} \), we are considering values of \( x \) that approach 1 from the left, i.e., \( x < 1 \). When \( x \) is less than 1, the expression \( 1 - x \) is positive, allowing us to evaluate the square root. Let's replace \( x \) with values slightly less than 1, like 0.9, or 0.99, and calculate \( \sqrt{1-x} \). This gives values approaching 0 as \( x \) approaches 1.

Step 2: Calculate One-Sided Limit A

As \( x \) approaches 1 from the left, \( 1-x \) approaches 0 from the positive side. Then, \( \sqrt{1-x} \) approaches 0. Therefore, \( \lim_{x \rightarrow 1^{-}} \sqrt{1-x} = 0 \).

Step 3: Understand One-Sided Limit B

For the limit \( \lim_{x \rightarrow 1^{+}} \sqrt{1-x} \), we consider values of \( x \) that are greater than 1. In this case, \( 1-x \) becomes negative, and the square root of a negative number is undefined in the real numbers. Hence, the expression doesn't yield a real number as \( x \) approaches 1 from the right.

Step 4: Determine One-Sided Limit B

Since \( \sqrt{1-x} \) is undefined for \( x > 1 \) in the real numbers, the limit \( \lim_{x \rightarrow 1^{+}} \sqrt{1-x} \) does not exist (in the real number system).

Step 5: Evaluate Two-Sided Limit

To find \( \lim_{x \rightarrow 1} \sqrt{1-x} \), check both one-sided limits. Since \( \lim_{x \rightarrow 1^{-}} \sqrt{1-x} = 0 \) but \( \lim_{x \rightarrow 1^{+}} \) does not exist in the real numbers, \( \lim_{x \rightarrow 1} \sqrt{1-x} \) does not exist because the limits from the two sides approach different "states" (existing vs. non-existing).

Key Concepts

Understanding One-Sided Limits

When evaluating limits, one-sided limits are essential for understanding how a function behaves as it approaches a specific point from one side only. Let's break down this concept using the following points:

• Left-Sided Limit: The limit as \( x \) approaches a point from the left side, typically denoted as \( x \to c^- \). This involves finding the value the function approaches as \( x \) gets closer to \( c \) from values less than \( c \).


• Right-Sided Limit: The limit as \( x \) approaches a point from the right side, noted as \( x \to c^+ \). This examines what the function approaches as \( x \) nears \( c \) from values greater than \( c \).

To evaluate one-sided limits, you substitute values into the function that are close to the target from the designated side. This indicates how the function behaves and allows us to predict approaching values without touching the exact point. Using \( \lim _{x \to 1^-} \sqrt{1-x} \) as an example, we find the limit is 0, since the values under the square root remain positive and shrink to zero. This careful analysis on each side helps anticipate the behavior of functions without crossing undefined boundaries.

Dealing with Undefined Expressions

In calculus, some expressions become undefined under certain conditions. Understanding when and why this happens is crucial, especially in limit evaluation. Here are some situations where expressions are undefined:

• Square Roots of Negative Numbers: In the real number system, you cannot take the square root of a negative number, resulting in an undefined expression. This occurs in the situation \( \lim_{x \rightarrow 1^{+}} \sqrt{1-x} \). As \( x \) approaches 1 from the right, \( 1-x \) becomes negative, making the square root function undefined in real numbers.


• Division by Zero: Another common source of undefined expressions, because division by zero is not valid. Approaching a limit that leads to such a scenario needs careful reevaluation or reformulation of the problem.

Recognizing undefined expressions helps us determine if a limit exists or not. This insight can guide us in deciding when a limit results in a real value and when further steps or different methods are required.

Real Number System Insights

The real number system is the foundation of many calculus problems, including limits. It's vital for understanding where expressions are valid and where they aren't. Let's break down some important aspects:

• Real Numbers: Include all rational and irrational numbers that can be represented on the number line, excluding the imaginary numbers. They are crucial in limit problems involving continuous functions and well-defined intervals.


• Constraints of Real Numbers: Within real numbers, certain operations are restricted. Notably, the square root function requires non-negative operands to yield real numbers. Hence, when \( x > 1 \), \( \sqrt{1-x} \) becomes invalid within the real numbers.

The real number system helps in framing the context in which we evaluate limits. It delineates the boundaries of valid expression and helps avoid pitfalls with undefined operations.