Problem 40 - A Graphical Approach to Precalculus with Limits

Question

Determine each limit, if it exists. $$\lim _{x \rightarrow 4}\left[\log _{2}(14+\sqrt{x})\right]$$

Step-by-Step Solution

Answer

The limit is 4.

Step 1: Identify the Limit Expression

The expression given is \( \lim_{x \rightarrow 4} \log_{2}(14+\sqrt{x}) \). We are tasked with finding the limit of this function as \( x \) approaches 4.

Step 2: Rewrite the Expression at the Limit Point

Substitute \( x = 4 \) directly into the function: \( \log_{2}(14+\sqrt{4}) = \log_{2}(14+2) = \log_{2}(16) \).

Step 3: Evaluate the Logarithm

Simplify \( \log_{2}(16) \). Since \( 2^{4} = 16 \), we know that \( \log_{2}(16) = 4 \).

Step 4: Conclude the Limit

Since substituting directly into the function and evaluating gives us a defined numerical value, we confirm the limit exists and is equal to this value.

Key Concepts

Logarithmic Functions

Logarithmic functions, often noted as \( \log_b(x) \), are the inverses of exponential functions. This means they answer the question: "To what power must we raise \( b \) (the base) to obtain \( x \)?". For example, \( \log_2(16) = 4 \) because \( 2^4 = 16 \).

When evaluating logarithmic expressions in limits, it's important to understand how the base and the number inside the logarithmic function interact. By simplifying the number inside the logarithmic function, or substituting values directly into it, we unsettle and solve the expression to find the power or exponent.

• Base of Logarithm: It is the number that is raised to a power.
• Input to Logarithm: This can be any positive number and it needs to be computable into a form of the base.
• Logarithm Output: This is the exponent, the answer to the logarithmic question.

Understanding the properties of logarithms, like \( \log_{b}(xy) = \log_{b}x + \log_{b}y \) and \( \log_{b}\left(\frac{x}{y}\right) = \log_{b}x - \log_{b}y \), helps make complex expressions manageable.

Square Roots

Square roots are a crucial part of many calculus problems and are denoted as \( \sqrt{x} \). They represent a number which, when multiplied by itself, yields \( x \). In simpler terms, if \( a^2 = x \), then \( a = \sqrt{x} \). For example, \( \sqrt{16} = 4 \) because \( 4^2 = 16 \).

In the context of evaluating limits, square roots often appear in expressions that require simplification before a limit can directly be determined. Often, direct substitution is the simplest approach. However, when that's not feasible due to indeterminant forms or other complexities, further algebraic manipulation may be necessary. Remember these points:

• Operation of Square Roots: Simply reduces a number to its root form.
• Factorization: Useful if expressions inside the square root are complex.
• Direct Substitution: Replace \( x \) with the approaching value.

These concepts make it easier to handle and simplify problems involving square roots and limits.

Evaluating Limits

Evaluating limits is the process of determining the value that a function approaches as the input approaches some value. When given an expression such as \( \lim_{x \to c} f(x) \), your task is to find out the value of the function as \( x \) approaches \( c \).

There are several methods to evaluate limits:

• Direct Substitution: Simply plug in the value of \( x \) into the function, as long as the function remains defined at that point.
• Simplification: This might involve factoring, rationalizing, or expanding functions to eliminate any undefined terms.
• Limits of Composite Functions: When functions are nested, you'll often address them one at a time, simplifying wherever possible.

Is crucial to determine whether substituting x directly results in an expression that can be computed without causing any mathematical anomalies, like division by zero. If so, the limit exists and equals the computed value.