Problem 7 - A Graphical Approach to Precalculus with Limits

Question

Let \(\lim _{x \rightarrow 4} f(x)=16\) and \(\lim _{x \rightarrow 4} g(x)=8 .\) Use the limit rules to find each limit. Do not use a calculator. $$\lim _{x \rightarrow 4} 2^{g(x)}$$

Step-by-Step Solution

Answer

The limit is 256.

Step 1: Understand Limit Properties

When calculating limits, we can use the property that the limit of an exponential function can be found by substituting the limit of the exponent. Thus, the limit of an exponential function like \( 2^{g(x)} \) can be expressed as \( 2^{\lim_{x \to 4} g(x)} \) if the limit of \( g(x) \) exists.

Step 2: Substitute Known Limits

From the given information, we know \( \lim_{x \to 4} g(x) = 8 \). Using the property identified in Step 1, substitute this into the exponential function: \[ \lim_{x \to 4} 2^{g(x)} = 2^{\lim_{x \to 4} g(x)} = 2^8. \]

Step 3: Calculate the Exponential Expression

Now that we have established that \( \lim_{x \to 4} 2^{g(x)} = 2^8 \), we can calculate the value of \( 2^8 \). This is equal to \( 256 \).

Key Concepts

Exponential Functions

Exponential functions are a type of mathematical function where a constant base is raised to a variable exponent. They are generally represented in the form \( a^x \), where \( a \) is a positive constant and \( x \) is the exponent, which could be a number or another expression.

Key characteristics of exponential functions include:

• They grow very rapidly compared to linear or polynomial functions. For instance, \( 2^x \) will increase much faster than \( x^2 \) as \( x \) becomes larger.
• The function value increases continuously and without bound if \( a > 1 \).
• If \( 0 < a < 1 \), the function decreases or decays as \( x \) increases.

In our exercise, we dealt with an exponential function of the form \( 2^{g(x)} \) at \( x = 4 \). Understanding the behavior of this function is crucial, as the output is highly dependent on the value of \( g(x) \) when \( x \) approaches 4.

Limit Properties

Limit properties are rules used to simplify the process of finding the limits of complex expressions. They involve operations such as addition, subtraction, multiplication, division, and exponentiation.

One fundamental limit property is that the limit of a function raised to a power is the power of the limit of the function itself. This is specifically useful in our context where we analyze \( 2^{g(x)} \). We can express this as \( 2^{\lim_{x \to c} g(x)} \) if the limit \( \lim_{x \to c} g(x) \) exists.

For our example, since \( \lim_{x \to 4} g(x) = 8 \), it follows that \( \lim_{x \to 4} 2^{g(x)} = 2^8 \). Understanding how these properties work allows us to solve limits without delving into complex calculations, emphasizing logic over computational effort.

Calculus

Calculus is a branch of mathematics focusing on limits, functions, derivatives, integrals, and infinite series. It provides tools for modeling and analyzing change, which is essential in fields like physics, engineering, economics, and beyond.

In calculus, limits are foundational. They help us understand the behavior of functions as they approach specific values. In the context of limits, we often deal with questions such as "What value does a function approach as the variable gets infinitely close to a certain point?"

This exercise shows the power of calculus by using limits to solve for exponential expressions without needing direct calculation. By understanding the properties of limits and exponential functions, calculus simplifies complex scenarios into manageable problems. This allows us to find exact values like "\( \lim_{x \to 4} 2^{g(x)} \)" smoothly using theoretical insights rather than numerical computation.