Problem 8 - The Calculus with Analytic Geometry

Question

Find the limits, and when applicable indicate the limit theorems being used. $$ \lim _{x \rightarrow+\infty} \frac{3 x^{4}-7 x^{2}+2}{2 x^{4}+1} $$

Step-by-Step Solution

Answer

The limit is \(\frac{3}{2}\).

Step 1: Identify the leading terms

The leading terms in the numerator and the denominator are the terms with the highest powers of x. For the numerator, the leading term is \(3x^4\), and for the denominator, the leading term is \(2x^4\).

Step 2: Simplify the expression

We can simplify the limit by dividing both the numerator and the denominator by \(x^4\). This gives us: \[\frac{3 - \frac{7}{x^2} + \frac{2}{x^4}}{2 + \frac{1}{x^4}}\].

Step 3: Evaluate the limit as x approaches infinity

As \(x\) approaches infinity, the terms \(\frac{7}{x^2}\), \(\frac{2}{x^4}\), and \(\frac{1}{x^4}\) approach zero. This simplifies the expression to: \[\frac{3 - 0 + 0}{2 + 0} = \frac{3}{2}\].

Step 4: Apply the limit theorem (if needed)

By applying the limit theorem for rational functions, since the degrees of the numerator and the denominator are equal, the limit is given by the ratio of the leading coefficients. Therefore, the limit is: \(\frac{3}{2}\).

Key Concepts

Introduction to Limit Theorems

Limit theorems are fundamental tools in calculus. They help us to find the limit of a function as the input approaches a certain value.

These theorems can be applied to various types of functions, including rational functions. Some key limit theorems include the limit laws for sums, differences, products, and quotients of functions.

For example:

• Sum Rule: \(\begin{vmatrix} \text{If } \lim_{x \to c} f(x) = L \text{ and } \lim_{x \to c} g(x) = M, \text{then } \lim_{x \to c} [f(x) + g(x)] = L + M \end{vmatrix} \).


• Product Rule: \(\begin{vmatrix} \text{If } \lim_{x \to c} f(x) = L \text{ and } \lim_{x \to c} g(x) = M, \text{then } \lim_{x \to c} [f(x) \cdot g(x)] = L \cdot M \end{vmatrix} \).

Limit theorems simplify complex limits by breaking down each part. This approach is crucial for understanding the behavior of rational functions as they approach infinity or another critical value.

Understanding Rational Functions

Rational functions are fractions where both the numerator and the denominator are polynomials. They can be expressed as:

• \(R(x) = \frac{P(x)}{Q(x)} \)

Where \(P(x) \) and \(Q(x) \) are polynomial functions of \(x\).

Rational functions are significant because they often arise in real-life problems and advanced fields like physics and engineering.

One common task is finding the limit of a rational function as \(x\rightarrow±∞\). To simplify this, we focus on the leading terms of their polynomials.

The Role of Leading Terms

Leading terms are the terms with the highest power of \(x\) in a polynomial.

For large values of \(x\), the leading terms dominate the behavior of the polynomial. Respectively, consider: \(P(x)=3x^4-7x^2+2\) and \(Q(x)=2x^4+1\). Here, \(3x^4\) is the leading term in the numerator, and \(2x^4\) in the denominator.
By simplifying the expression: \(\frac{3x^4-7x^2+2}{2x^4+1} \rightarrow \frac{3-\frac{7}{x^2}+\frac{2}{x^4}}{2+\frac{1}{x^4}}\), we can analyze the limit as \(x\rightarrow +∞\).
The term \(3x^4\) heavily influences the value, making the smaller terms insignificant. Thus, \(Lim_{x\rightarrow +∞} P(x)/Q(x)\) becomes the ratio of leading coefficients: \(3/2\). This approach effectively breaks complex problems into simpler, manageable parts.