Problem 11 - 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. $$\lim _{x \rightarrow-3^{-}} \sqrt{x+3}$$

Step-by-Step Solution

Answer

The limit does not exist because \( \sqrt{x+3} \) is undefined for \( x < -3 \).

Step 1: Understanding the Limit

We are tasked with finding the limit of the function \( f(x) = \sqrt{x+3} \) as \( x \) approaches \(-3^{-}\), which means from the left side. The limit process involves examining the behavior of the function as \( x \) gets arbitrarily close to \(-3\) from values less than \(-3\).

Step 2: Setting Up the Expression Inside the Square Root

For the function \( \sqrt{x+3} \), if \(x \) approaches \(-3\) from the left (i.e., \(x \to -3^-\)), then values of \( x+3 \) approach zero from the negative side. Therefore, we need to check what happens as the expression under the square root approaches zero.

Step 3: Analyzing the Square Root Function

The square root function, \( \sqrt{z} \), is defined for non-negative values of \( z \). As \( x+3 \to 0^-\), which means \( x+3 \) approaches zero from the negative side, \( \sqrt{x+3} \) becomes undefined because the square root of a negative number is not real (within the realm of real numbers).

Step 4: Concluding the Limit

Since \( \sqrt{x+3} \) is undefined for values where \(x+3 < 0\), the limit \( \lim _{x \rightarrow-3^{-}} \sqrt{x+3} \) does not exist. The square root function only yields real numbers when the radicand is positive or zero.

Key Concepts

Analyzing Graphs

When understanding limits in calculus, analyzing graphs is an incredibly useful skill. Graphs provide a visual representation of how functions behave as inputs approach certain values.For instance, a graph can show us the trend of a function, making it easier to grasp the behavior of the function as it approaches a specific point from the left or right.In the given problem, we deal with the limit \( \lim_{x \to -3^{-}} \sqrt{x+3} \).Analyzing the graph of the square-root function, we'd notice it is only defined for specific portions of the x-axis.
To solve the limit, we would observe how the points on the graph behave:

• If \(x+3\) gets close to zero from below, the graph will show that the function does not produce any real number output.
• The function's graph will not touch or cross into negative values for the input \(x+3 \).

This absence of values will visually confirm that the limit does not exist, as there is no real number result as \(x\) approaches -3 from the left side.

Square Root Function

The square root function is a fundamental concept often encountered in mathematics, including calculus. The square root of a number \( z \) is written as \( \sqrt{z} \) and is only defined for non-negative values of \( z \).This is because imaginary numbers come into play when taking square roots of negative numbers, which are outside the scope of real numbers.For the function \( \sqrt{x+3} \):

• It is essential to recognize that \( x+3 \) must be greater than or equal to zero for \( \sqrt{x+3} \) to produce real numbers.
• Negative values of \( x+3 \) make the expression \( \sqrt{x+3} \) undefined in the realm of real numbers.

In the context of limits, this means if the value inside the square root approaches zero from the negative side, the expression is undefined.Hence, the limit for the scenario \( \lim_{x \to -3^{-}} \sqrt{x+3} \) simply does not exist in the real number system.

Real Numbers

The concept of real numbers is fundamental to understanding limits in calculus, as well as various other mathematical concepts. Real numbers include all the numbers on the number line, including both rational and irrational numbers, covering positive, zero and negative values.
However, certain operations like taking the square root introduce limitations on the kinds of inputs we can use within the scope of real numbers.The square root of a negative number does not produce a real number, leading instead to complex numbers, which involve the imaginary unit \( i \).Therefore:

• In our given exercise, as \( x+3 \) approached zero from the left (negative values), the output for \( \sqrt{x+3} \) cannot be part of the set of real numbers.
• This is why one needs to understand real numbers are constrained in certain operations, affecting the outcomes of limits and existence within such operations.

Grasping the properties of real numbers can greatly aid in understanding when limits will or will not exist, particularly when dealing with square roots.