Problem 34 - Calculus

Question

For \(f(x)=6 x^{3}+1000 x\) find (a) \(\lim _{x \rightarrow \infty} \frac{f(x)}{x}\) (b) \(\lim _{x \rightarrow \infty} \frac{f(x)}{x^{2}}\) (c) \(\lim _{x \rightarrow \infty} \frac{f(x)}{x^{4}}\) (d) \(\lim _{x \rightarrow \infty} \frac{f(x)}{x^{3}+1}\)

Step-by-Step Solution

Answer

(a) ∞, (b) ∞, (c) 0, (d) 6

Step 1: Simplify the Expression for (a)

To find \( \lim_{x \rightarrow \infty} \frac{f(x)}{x} \), substitute \( f(x) = 6x^{3} + 1000x \) into the expression:\[\lim_{x \rightarrow \infty} \frac{6x^{3} + 1000x}{x} = \lim_{x \rightarrow \infty} \left( 6x^{2} + 1000 \right)\]As \(x\) approaches infinity, the \(6x^{2}\) term will dominate, leading to the limit approaching infinity.

Step 2: Simplify the Expression for (b)

To find \( \lim_{x \rightarrow \infty} \frac{f(x)}{x^{2}} \), substitute \( f(x) = 6x^{3} + 1000x \) into the expression:\[\lim_{x \rightarrow \infty} \frac{6x^{3} + 1000x}{x^{2}} = \lim_{x \rightarrow \infty} \left( 6x + \frac{1000}{x} \right)\]As \(x\) approaches infinity, the \(6x\) term will dominate since \(\frac{1000}{x}\) approaches zero, resulting in the limit going to infinity.

Step 3: Simplify the Expression for (c)

To find \( \lim_{x \rightarrow \infty} \frac{f(x)}{x^{4}} \), substitute \( f(x) = 6x^{3} + 1000x \) into the expression:\[\lim_{x \rightarrow \infty} \frac{6x^{3} + 1000x}{x^{4}} = \lim_{x \rightarrow \infty} \left( \frac{6x^{3}}{x^{4}} + \frac{1000x}{x^{4}} \right)\]This becomes:\[\lim_{x \rightarrow \infty} \left( \frac{6}{x} + \frac{1000}{x^{3}} \right)\]As \(x\) approaches infinity, both terms approach zero, resulting in the limit of zero.

Step 4: Simplify the Expression for (d)

To find \( \lim_{x \rightarrow \infty} \frac{f(x)}{x^{3} + 1} \), substitute \( f(x) = 6x^{3} + 1000x \) into the expression:\[\lim_{x \rightarrow \infty} \frac{6x^{3} + 1000x}{x^{3} + 1} = \lim_{x \rightarrow \infty} \left( \frac{6x^{3}}{x^{3} + 1} + \frac{1000x}{x^{3} + 1} \right)\]Focus on the dominant term:\[\lim_{x \rightarrow \infty} \left( \frac{6x^{3}}{x^{3} + 1} \right)\]Since \(\frac{6x^{3}}{x^{3} + 1} \approx 6\) when \(x\) approaches infinity, as the \(+1\) becomes negligible, the limit is 6.

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions involving sums of powers of variables. The general form of a polynomial function is: \[ f(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \] where \(a_n, a_{n-1}, \ldots, a_0\) are constants called coefficients, and \(n\) is a non-negative integer.

• The highest power of the variable \(x\) in the polynomial determines its degree.
• The leading term is the term with the highest power, such as \(6x^3\) in \(f(x) = 6x^3 + 1000x\).

Using the leading term, you can often predict the behavior of the polynomial as \(x\) grows very large or very small. Polynomial functions are continuous and smooth, providing a great tool for modeling and exploring complex real-world situations.

Limit at Infinity

The concept of a limit at infinity helps us understand the behavior of functions as the input approaches infinity (or negative infinity). When evaluating limits at infinity:

• We focus on how a function behaves as \(x\) becomes very large or very large negatively.
• The dominant term (the one with the highest power when \(x > 0\)) dictates the limit's value, especially with polynomial functions.

For example, in evaluating \(\lim_{x \rightarrow \infty} \frac{f(x)}{x^2}\), the function simplifies thanks to the dominant term, allowing us to predict behavior without actual calculation. Here, the limit is driven by the behavior of \(6x\) as it goes to infinity.

Dominant Term

The dominant term is crucial in understanding limits involving polynomial functions. It is the term in the polynomial with the highest degree. As \(x\) becomes large, the effect of all lesser degree terms fades in comparison. This is because higher powers of \(x\) grow faster than lower powers:

• In \(6x^3 + 1000x\), the dominant term is \(6x^3\) because it has the highest exponent (3).
• As \(x\) approaches infinity, \(6x^3\) vastly outweighs the impact of \(1000x\).

This knowledge simplifies solving limit problems as you can often predict outcomes by considering just the dominant term's behavior.

Infinite Limits

Infinite limits describe a situation where the limit of a function becomes infinitely large as \(x\) approaches a certain value or increases indefinitely. Infinite limits can occur in polynomial functions when:

• The numerator, such as in \(6x^3\), increases without bound faster than the denominator, leading to an infinite result.
• Both numerator and denominator increase, but the numerator does so much faster, as seen in \(\lim_{x \rightarrow \infty} \frac{6x + \frac{1000}{x}}{1}\).

Understanding infinite limits helps in determining how graphs of functions behave at their extremities, providing a clearer picture of overall function behavior at large scales.