Problem 28 - Calculus Early Transcendentals

Question

Limits of compositions Evaluate each limit and justify your answer. $$\lim _{x \rightarrow 2}\left(\frac{3}{2 x^{5}-4 x^{2}-50}\right)^{4}$$

Step-by-Step Solution

Answer

Question: Evaluate the limit: $$\lim _{x \rightarrow 2}\left(\frac{3}{2 x^{5}-4 x^{2}-50}\right)^{4}$$ Answer: The limit exists, and its value is \(\frac{81}{26214400}\).

Step 1: Analyze the inner function

Let's first analyze the function within the parentheses, \(f(x) = \frac{3}{2 x^{5} - 4 x^{2} - 50}\). Substitute the value of x = 2 into the function: \(f(2) = \frac{3}{2(2)^{5} - 4(2)^{2} - 50} = \frac{3}{80}\) Since the inner function is defined at x=2, we can proceed with finding the limit of the entire expression.

Step 2: Evaluate the limit

Since the inner function is defined at x = 2, we can use substitution to find the limit of the entire expression: $$\lim _{x \rightarrow 2}\left(\frac{3}{2 x^{5}-4 x^{2}-50}\right)^{4} = \left(\frac{3}{80}\right)^{4}$$

Step 3: Simplify the result

Now, raise the fraction to the power of 4: $$\left(\frac{3}{80}\right)^{4} = \frac{3^4}{80^4} = \frac{81}{26214400}$$

Step 4: Justify the answer

By substitution, we determined that the inner function was defined at x = 2. Consequently, the entire expression has a limit, and we were able to derive the exact value of the limit: $$\lim _{x \rightarrow 2}\left(\frac{3}{2 x^{5}-4 x^{2}-50}\right)^{4} = \frac{81}{26214400}$$ The limit exists, and its value is \(\frac{81}{26214400}\).

Key Concepts

Composite Functions

In calculus, a composite function is an expression where one function is applied inside another. You can think of a composite function as a chain of functions. Each function affects the next one in sequence. When dealing with limits, however, you'll focus on the inner function first. The composite function given here is

• The inner part: \( f(x) = \frac{3}{2x^5 - 4x^2 - 50}\)
• The outer part: raising the result to the power of four, which is \( ext{outer function} = ( ext{inner function})^4\)

This nested structure means that before tackling the overall limit, you analyze and evaluate the inner function. Once you know its limit at a particular point, you use that result in the outer function.

Polynomial Functions

A polynomial function is a mathematical expression that involves sums and powers of variables, often denoted with x. The inner function we are examining, \( 2x^5 - 4x^2 - 50 \), is a polynomial. Polynomial functions are particularly convenient because they are continuous and defined for all real numbers, which simplifies finding limits.

• The degree of the polynomial is determined by the highest power of x, which here is 5.
• Coefficients in front of the terms, like 2 in \( 2x^5\), influence the graph's stretching.

Understanding this helps when you substitute in a specific value of x, as you can directly calculate the polynomial's result. Here, evaluating this function at x = 2 ensures the expression is defined, enabling us to explore the limit further.

Substitution Method

The substitution method in limits involves directly replacing the variable with a specific value. This makes finding limits straightforward for functions that are continuous at the point of interest. Here, using the substitution method, we replaced

• Replace x with 2 in the inner function, \( rac{3}{2(2)^5 - 4(2)^2 - 50}\), and got \( rac{3}{80}\).
• Then, by raising this result to the power of 4, we evaluated the composite function's limit.

If a function isn't continuous, substitution might not properly reflect the limit. However, in many polynomial cases, like the one in our problem, substitution provides an efficient route to the answer, culminating in the final limit result: \( \frac{81}{26214400}\). This showcases how substitution aids in locating a precise limit value when the function behaves predictably.