Problem 48 - 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. $$\lim _{x \rightarrow 3^{+}}\left[x^{2}+2 x\right]$$

Step-by-Step Solution

Answer

The right-hand limit is 15.

Step 1: Understanding One-Sided Limits

The notation \( \lim _{x \rightarrow 3^{+}}\left[x^{2}+2 x\right] \) refers to the right-hand limit as \( x \) approaches 3. This means we're considering values of \( x \) that are just a bit greater than 3.

Step 2: Substitute the Limit Value

To find the right-hand limit, plug \( x = 3 \) into the expression \( x^2 + 2x \). Thus, you calculate: \( 3^2 + 2 \cdot 3 \).

Step 3: Calculate the Expression

Now compute the expression: \( 3^2 = 9 \) and \( 2 \times 3 = 6 \). Then, sum them up to get \( 9 + 6 = 15 \).

Step 4: Conclude the Right-Hand Limit

Since there is no discontinuity within the values coming from the right of 3, the right-hand limit as \( x \rightarrow 3^{+} \) is equal to the expression value at \( x = 3 \). Therefore, the right-hand limit is \( 15 \).

Key Concepts

One-Sided Limits

One-sided limits are crucial in understanding how a function behaves as it approaches a particular point from one direction only. They are great tools when dealing with functions that may not be continuous everywhere. In mathematical terms, a one-sided limit could be from the right, symbolized by a "+" sign (right-hand limit), or from the left, symbolized by a "−" sign (left-hand limit).

Understanding one-sided limits allows us to analyze functions where behavior might differ depending on the path of approach. For example, when there might be a jump or a hole at a point in the function. This concept helps in exploring limits in a more detailed manner by choosing a perspective either gradually increasing or decreasing towards a point.

When you see notation like \( \lim_{x \to a^{+}} f(x) \), it signifies that values of \( x \) are slightly more than \( a \). Being aware of this ensures you evaluate the functions correctly when needed. Overall, recognizing one-sided limits can give a deeper understanding of the behavior of complex functions.

Limit Evaluation

To evaluate limits, especially one-sided limits, it's akin to predicting the output value as the input value approaches a specified point. In simple terms, limit evaluation helps find what a function is trending towards, rather than the function's actual value at that point.

Evaluating a limit generally involves substituting the approaching value into the function if it simplifies without causing undefined operations. Often, the steps include directly plugging in the approaching value, simplifying the expression, and observing the results. In cases where direct substitution causes division by zero or other indeterminate forms, alternative methods such as factoring or using conjugates might be necessary.

It's important to recognize the context of continuity and how it affects evaluation. If a function is continuous at the point, the limit can be directly calculated with ease. If the function has a break or jump, it might need separate evaluations for right-hand and left-hand limits.

Continuity

In calculus, continuity signifies that a function has no holes, jumps, or unexpected breaks at a point, allowing limits to approach smoothly from either direction. For a function \( f \) to be continuous at a point \( c \), the following must hold true:

• \( f(c) \) is defined.
• The limit \( \lim_{x \rightarrow c} f(x) \) exists.
• \( \lim_{x \rightarrow c} f(x) = f(c) \).

Continuity ensures that the limit from the left and the limit from the right are equal and equal to the function’s value at that point. If a function is continuous all throughout its domain, it makes limit evaluation straightforward, as there's no concern for discrepancies from different approaches. However, in its absence, examining each side separately through one-sided limits becomes indispensable.

Recognizing continuity helps in quickly solving limit problems and understanding functional behaviour extensively, ensuring there’s a smooth path across the interval.

Left-Hand Limit

When evaluating the left-hand limit, we focus on the behavior of a function as the input values approach a particular number from the left side. This is often denoted using the "−" sign in the limit notation, like \( \lim_{x \to a^{-}} f(x) \).

The process is quite similar to the right-hand limit but highlights the approach from smaller values than the target number. It's an invaluable method, especially when dealing with functions with asymmetry or discontinuities at points.

To find a left-hand limit, substitute values into the function that are slightly less than the target number and observe the resulting outputs. Watch out for functions with different expressions or rules when values are approached from different sides.

Understanding left-hand limits completes the picture of how a function behaves near a point. Therefore, it is equally important as right-hand limits when analyzing the overall behavior and continuity of functions.