Problem 32 - Calculus and its applications

Question

Use the Limit Properties to find the following limits. If a limit does not exist, state that fact. $$ \lim _{x \rightarrow 4} \sqrt{x^{2}-9} $$

Step-by-Step Solution

Answer

The limit is \( \sqrt{7} \).

Step 1: Identify the Limit Expression

The expression we need to find the limit for is \( \sqrt{x^2 - 9} \) as \( x \to 4 \).

Step 2: Check Direct Substitution Eligibility

To check whether we can directly substitute \( x = 4 \) into \( \sqrt{x^2 - 9} \), evaluate \( 4^2 - 9 \). This results in \( 16 - 9 = 7 \), which is a positive number.

Step 3: Apply Direct Substitution

Since \( x^2 - 9 = 7 > 0 \), we can directly substitute \( x = 4 \) into the expression. Therefore, the limit is \( \sqrt{4^2 - 9} = \sqrt{7} \).

Step 4: State the Result

After substitution, the expression evaluates to \( \sqrt{7} \). Hence, the limit exists and is equal to \( \sqrt{7} \).

Key Concepts

Direct Substitution

The method of direct substitution is streamlined and intuitive when finding limits. It entails replacing the variable approaching a specific value directly into the function. This technique is only feasible when substituting that value results in a well-defined number. For instance, for the expression \( \lim _{x \rightarrow 4} \sqrt{x^{2}-9} \), direct substitution means you insert \( x = 4 \) into \( \sqrt{x^{2}-9} \).
Before performing direct substitution, it's important to verify that the function does not become undefined or lead to indeterminate forms. Here, we compute \( 4^2 - 9 \), which results in 7. Because 7 is a definite, positive number, substitution results in a clear answer, \( \sqrt{7} \).
In cases where substitution would lead to undefined conditions like division by zero or negative roots under an even root (except for cube roots and such), the technique would require reassessment using other methods like factoring, rationalizing the numerator, or trigonometric limits.

Square Root Function

Square root functions involve expressions under the radical symbol, affecting how limits are handled. These functions are defined for non-negative values inside the root when dealing with real numbers.
For example, in \( \lim _{x \rightarrow 4} \sqrt{x^{2}-9} \), you first determine that the expression under the square root, \( x^2 - 9 \), results in a non-negative number at the \( x \) value of interest, ensuring legitimacy.
Key points about square root functions are:

• They are only valid for non-negative inputs when dealing with real numbers.
• Roots of negative numbers require complex numbers, not suited for basic calculus contexts regarding real limits.
• Ensure the radicand is non-negative for substitution to provide real results—here, \( x^2 - 9 = 7 \) is positive, confirming \( \sqrt{7} \) is achievable.

Existence of Limits

Determining the existence of limits is crucial before declaring any findings. A limit exists when, as the variable approaches a certain point, the function approaches a definite value.
In our given problem, \( \lim _{x \rightarrow 4} \sqrt{x^{2}-9} \), a limit exists because substituting \( x = 4 \) results in a clear value, \( \sqrt{7} \), fulfilling the condition for limit existence.
Remember these essentials for the existence of limits:

• The function must approach the same value from the left and right sides of the point of interest.
• If direct substitution yields an indeterminate form, further analysis like factoring or conjugates may be needed.
• Non-existent limits often signal undefined expressions or infinite behaviors, neither of which occur here thanks to our well-behaved square root function.