Problem 44 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find each of the right-hand and left-hand limits or state that they do not exist. $$\lim _{x \rightarrow 1^{-}} \frac{\sqrt{1+x}}{4+4 x}$$

Step-by-Step Solution

Answer

The left-hand limit is \( \frac{\sqrt{2}}{8} \).

Step 1: Understanding the Problem

We need to find the left-hand limit of the function \( \frac{\sqrt{1+x}}{4+4x} \) as \( x \) approaches 1 from the left. This means we are considering values of \( x \) that are slightly less than 1.

Step 2: Substitute Values Approaching from the Left

Substitute values that are slightly less than 1 (like 0.9, 0.99, etc.) into the function \( \frac{\sqrt{1+x}}{4+4x} \) and observe the outputs. This helps determine the behavior of the function as \( x \rightarrow 1^{-} \).

Step 3: Simplify and Evaluate

Notice that the function is continuous around \( x = 1 \) because both the numerator \( \sqrt{1+x} \) and the denominator \( 4+4x \) are continuous functions. Simplify: \( \frac{\sqrt{1+x}}{4+4x} \rightarrow \frac{\sqrt{1+1}}{4+4 \cdot 1} = \frac{\sqrt{2}}{8} \).

Step 4: Conclude the Left-Hand Limit

Since there are no points of discontinuity around \( x = 1 \), the limit as \( x \rightarrow 1^{-} \) simply evaluates to \( \frac{\sqrt{2}}{8} \). Thus, the left-hand limit exists and equals \( \frac{\sqrt{2}}{8} \).

Key Concepts

Right-hand limit

Right-hand limit is a fascinating concept in calculus, used to understand the behavior of a function as the input approaches a certain value from the right side. Think of it like slowly inching closer to a number from values that are just a bit bigger than it. To find the right-hand limit, we look at the values that are slightly greater than the point of interest. For a function, say \( f(x) \), if you want to find \( \lim_{x \to a^+} f(x) \), it implies viewing how \( f(x) \) behaves as \( x \) approaches \( a \) from numbers just larger than \( a \). This is crucial in identifying if the function behaves nicely without breaking or jumping as it crosses \( a \).

• Evaluate for numbers slightly greater than \( a \).
• Notice behavior as \( x \) approaches \( a \) from the right.

Being patient and observing this change points to understanding complex functions in calculus. This concept is critical in comprehending overall function behavior and continuity.

Left-hand limit

The left-hand limit is similar to a detective's magnifying glass that zooms in on the behavior of a function as it approaches a specific point from the left. This time, instead of arriving from the right, you're coming from a teeny bit smaller number.When evaluating \( \lim_{x \to a^-} f(x) \), you're watching \( f(x) \) as \( x \) creeps towards \( a \) but from the left. These left-side estimations tell us if the function gently moves to the point or if there are any abrupt jumps or unexpected shifts.

• Look at numbers less than \( a \) moving towards \( a \).
• Evaluate how \( f(x) \) behaves from the left side.

Mastering the left-hand limit ensures you grasp how a function trends towards a particular value—something that is exceptionally useful when solving limits like the example provided.

Continuity

Continuity in a function is akin to a smooth road without sudden turns or breaks. Formally, a function is continuous at a point if it doesn’t suddenly jump, that is, both the left-hand limit and right-hand limit exist and are equal to the function's value at that point.Imagine a function \( f(x) \) that is continuous at \( x = a \). For this to hold true:

• The limit from the left \( \lim_{x \to a^-} f(x) \) should exist.
• The limit from the right \( \lim_{x \to a^+} f(x) \) should also exist.
• Both these limits should equal \( f(a) \).

If all these match, the graph of the function will have no gaps or holes at \( x = a \). Such continuity assures that a function can be smoothly drawn at that point, which is crucial in many real-world applications where predictability and steady behavior are needed.

Evaluating limits

Evaluating limits is a cornerstone of calculus, acting as a primary tool to explore function behaviors at points that might not be directly accessible—like approaching a steep hill or a cliff edge.To evaluate limits, consider both side approaches to the number of interest. Different methods can be employed:

• Direct Substitution: Plugging in the number directly and observing the outcome if the function remains intact without discontinuities.
• Factoring or Rationalizing: Simplifying expressions to eliminate uncertainties or indeterminate forms.
• Using L'Hôpital's Rule: When encountering forms like \( \frac{0}{0} \), derivatives may provide a clearer path to the limit.

Evaluating limits not only reveals the behavior of functions but also plays a crucial role in understanding derivatives and integrals—further expanding the horizon of mathematical exploration.