Problem 25 - Contemporary Precalculus

Question

Find the limit if it exists. If the limit does not exist, explain why. $$\lim _{x \rightarrow-25^{+}}(\sqrt{5+2 x}+x)$$

Step-by-Step Solution

Answer

Answer: The limit does not exist. The reason is that the function inside the square root is not defined for values close to -25 from the right side.

Step 1: Determine if the function is well defined for values close to -25 from the right side

Let's first look at the function inside the square root: \(5 + 2x\). As x approaches -25 from the right side, \(5 + 2x\) becomes more and more negative, which means it is undefined for values close to -25. Since the function inside the square root is undefined for those values, the limit doesn't exist for the given function. Thus, the answer is: The limit does not exist. The reason is that the function inside the square root is not defined for values close to -25 from the right side.

Step 2: Attempt direct substitution

Try substituting the value the variable approaches directly into the expression.

Step 3: Handle indeterminate forms

If direct substitution gives an indeterminate form, apply L'Hopital's Rule, algebraic manipulation, or other techniques.

Step 4: Evaluate the limit

Compute the final value of the limit.

Step 5: State the conclusion

Express the final answer.

Step 6: Conclude with the answer

Answer: The limit does not exist. The reason is that the function inside the square root is not defined for values close to -25 from the right side.

Key Concepts

Right-Hand Limits

When we talk about right-hand limits, we are focusing on the values that a function approaches as the input approaches a particular point from the right side. Consider the point where the limit is evaluated, say at \(x = a\). For a right-hand limit, this would mean that \(x\) is approaching \(a\) from values greater than \(a\) (i.e., \(a^{+}\)).

In the context of the exercise, we are looking at \(x \rightarrow -25^{+}\). This means \(x\) is getting very close to -25 while being slightly greater than -25, effectively inching towards it from the right.

Understanding right-hand limits helps in evaluating the behavior of functions near critical points, sometimes revealing differences in continuity or other aspects. Importantly, these limits might have different results than considering the limit from the left or considering a two-sided limit. Always consider right-hand limits carefully in conjunction with other types of limits to fully understand the function's behavior.

Square Roots in Limits

Square roots can be tricky when dealing with limits, especially if the expression inside the square root can become negative. The square root function \(\sqrt{x}\) is defined only for non-negative values of \(x\). This is because you can't take the real square root of a negative number in the real number system.

In the exercise, the expression under the square root is \(5 + 2x\). As \(x\) approaches -25 from the right, the value of \(5 + 2x\) decreases. If at any point it becomes negative, the function \(\sqrt{5 + 2x}\) becomes undefined for real numbers.

• If \(5 + 2x\) becomes negative, you can't find a real value for the square root.
• This can lead to the conclusion that the entire function becomes undefined.

Understanding how square roots behave in limits is crucial, especially when determining if a limit exists or not. Keep an eye on the domain of the function and remember that roots need non-negative inputs for square roots to exist in real numbers.

Function Undefined at Limit

Sometimes, while evaluating limits, a function may become undefined at or near a particular point. This typically occurs either due to division by zero, taking the square root of a negative number, or other operations that are not defined for some values.

In our exercise, the function becomes problematic as \(x\) approaches -25 from the right. This is primarily because the expression \(5 + 2x\) yields negative values, making the square root undefined. As a result, the entire limit does not exist—there are simply no outputs the function can provide.

• When a function is undefined at a limit, this suggests that no value can be determined as an output as \(x\) approaches the point from the specified side.
• In this case, we announced that the limit does not exist because a significant component of the function failed locally near the limit point.

Understanding these scenarios in limits helps diagnose problems in function behavior and continuity, emphasizing the importance of checking for undefined values during limit evaluation.