Problem 12 - A Graphical Approach to Precalculus with Limits

Question

Determine each limit. Refer to the accompanying graph of \(y=f(x)\) when it is given. Do not use a calculator. $$\lim _{x \rightarrow 2^{-}} \sqrt{2-x}$$

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Understand the Limit Notation

The expression \( \lim_{x \to 2^{-}} \sqrt{2-x} \) asks us to find the value that the function \( \sqrt{2-x} \) approaches as \( x \) approaches 2 from the left side (denoted by \( 2^{-} \)).

Step 2: Analyze the Expression as x Approaches 2 from the Left

Since we are approaching 2 from the left, \( x \) is slightly less than 2. Set \( x = 2-h \) where \( h \to 0^{+} \). Then, \( 2-x = 2-(2-h) = h \), which is a very small positive number.

Step 3: Evaluate the Limit

Substitute the expression from Step 2 into the function: \( \sqrt{2-x} = \sqrt{h} \). As \( h \to 0^{+} \), \( \sqrt{h} \to \sqrt{0} = 0 \).

Step 4: Confirm the Limit Value

We observe that as \( x \to 2^{-} \), the function \( \sqrt{2-x} \) indeed approaches 0. Therefore, the limit is 0.

Key Concepts

Left-Hand Limit

Limits play a crucial role in understanding calculus, particularly when analyzing how functions behave near specific points. A helpful tool in these analyses is the concept of **left-hand limits**. This refers to the value a function approaches as the input variable approaches a specified point from the left side.In notation, a left-hand limit is often expressed with a minus sign in the exponent of the point being approached. For instance, in the expression \( \lim_{x \to 2^{-}} \sqrt{2-x} \), the \( 2^{-} \) indicates we are considering the behavior of the function '\( \sqrt{2-x} \)' as \( x \) approaches 2 from values less than 2.When evaluating left-hand limits:

• Identify the direction from which \( x \) approaches the point. In this case, from the left.
• Convert \( x \) into a form that reflects its closeness to the point, like \( x = 2 - h \), where \( h \) is a small positive number going to zero.

Understanding this approach allows us to predict how functions act without error from graphical interpretations or calculator misuse.

Function Behavior

Function behavior describes how a function acts as its input variable nears a particular point. Observing this behavior helps us to understand more than just the outcome at a specific value—it illuminates the overarching trends as we approach that point.Consider the function \( \sqrt{2-x} \) as \( x \) approaches 2 from the left, or \( 2^{-} \). Below is how to analyze its behavior:

• As \( x \to 2^{-} \), substitute \( x = 2 - h \), with \( h \to 0^{+} \).
• The expression simplifies to \( \sqrt{h} \), a square root of an increasingly small positive number.
• Thus, the function’s value decreases toward zero as \( x \) nears 2 from the left.

This analysis highlights that as one moves closer to the point on the graph, the values of \( \sqrt{2-x} \) approach zero, showing clear function behavior tendencies.

Approaching a Point

In limits, "approaching a point" implies examining how the values of a function are trending as the input variable gets closer to a specific value. It does not necessarily require evaluating the function at that point itself.When examining \( \lim_{x \to 2^{-}} \sqrt{2-x} \), we are interested in how the function behaves as \( x \) nears 2 from the left-hand side:

• As \( x \) gets closer to 2, \( 2-x \) results in smaller positive values.
• When \( x = 2 - h \), with \( h \) a small positive value heading to zero, \( \sqrt{2-x} \) translates to \( \sqrt{h} \).
• The function values are trending towards zero, not at 2 exactly, but as they get infinitely close to it.

By understanding how the function approaches a point, students can anticipate and describe the limit with greater clarity.