Question

Evaluate the given limit. $$ \lim _{x \rightarrow 3} 4^{x^{3}-8 x} $$

Short answer

The limit \( \lim_{x \to 3} 4^{x^3 - 8x} \) is 64.

Step-by-step solution

Recognize the type of limit

We are asked to evaluate \[ \lim_{x \to 3} 4^{x^3 - 8x}.\] This is a limit evaluation with an exponential function.

Simplify the exponent function

Simplify the exponent function inside the limit:\[ x^3 - 8x = x(x^2 - 8). \] Substitute \(x = 3\) into this function:\[ 3^3 - 8 \times 3 = 27 - 24 = 3. \] The exponent approaches 3.

Evaluate the limit of the base

Since we have simplified the exponent function, we need to evaluate the base of the exponential due to its constant value:\[ 4^3 = 64.\] The limit of the base remains 4.

Apply the Limit Laws

Through the limit evaluation of the exponential, we have:\[ \lim_{x \to 3} 4^{x^3 - 8x} = 4^3 = 64. \] The limit laws allow us to substitute directly since the function is continuous and the exponent approaches a constant.

Key concepts

Exponential Function

An exponential function has the form \( a^{f(x)} \), where \( a \) is a constant and \( f(x) \) is some function of \( x \). Exponential functions are critical in many areas of mathematics and science because they model growth and decay processes. For instance, compound interest or population growth can often be described by exponential functions.

In the exercise, we have \( 4^{x^3 - 8x} \). Here, 4 is the base, which is a positive constant, and the exponent is a polynomial function \( x^3 - 8x \). This specific form makes it particularly easy to evaluate since the exponential function remains continuous and differentiable, a property that simplifies finding limits.

To evaluate limits of exponential functions practically, you first look at the behavior of the exponent as the variable approaches a particular value. Understanding how the exponents change provides insight into how the entire expression behaves.

Continuous Function

A function is continuous if there are no "holes," gaps, or jumps in its graph. In simpler terms, you can draw its graph in one smooth motion without lifting your pen or pencil from the paper. Continuous functions are desirable because they imply predictability and reliability. Calculating limits of continuous functions can be straightforward because the function value at a point can be directly evaluated to find the limit.

In the context of the problem, \( x^3 - 8x \) is a polynomial, and all polynomials are continuous over all real numbers. This helps because it means that substituting \( x = 3 \) directly into this polynomial allows us to find the exponent at \( x = 3 \) without any additional considerations.

The base of our exponential function, namely 4, also maintains continuity since it is a constant. So, once the exponent is computed as described, the process of finding a limit becomes direct and seamless, thanks to this continuity.

Limit Laws

Limit laws are the foundational rules that allow for the evaluation of limits, making them easier to handle and break down. These laws include the sum, difference, product, and quotient rules, along with special considerations for polynomial, rational, and exponential functions.

In our problem, we took advantage of several limit laws. Crucially, because the exponential function and its exponent result in constants at \( x = 3 \), it implied that \[ \lim_{x \to 3} 4^{x^3 - 8x} = 4^3 = 64. \] This was possible due to the continuity of the functions involved and the limit law allowing the substitution of the function value at the point of interest.

Recognizing continuous exponential functions and applying these laws is key in simplifying complex expressions and evaluating limits efficiently. Providing that the limits of the components individually exist and align, the composite limit itself will reflect these results, as shown in the solution.