Problem 1 - Calculus for Biology and Medicine

Question

Evaluate the limits. $$ \lim _{x \rightarrow \infty} \frac{2 x^{2}-3 x+5}{x^{4}-2 x+1} $$

Step-by-Step Solution

Answer

The limit as \( x \to \infty \) is 0.

Step 1: Identify Leading Terms

To evaluate the limit of a rational function as \( x \to \infty \), identify the leading terms in the numerator and the denominator. The leading term in the numerator is \( 2x^2 \), and the leading term in the denominator is \( x^4 \).

Step 2: Simplify the Expression

Divide both the numerator and the denominator by \( x^4 \), the highest power of \( x \) present. This gives us: \[\frac{\frac{2x^2}{x^4} - \frac{3x}{x^4} + \frac{5}{x^4}}{\frac{x^4}{x^4} - \frac{2x}{x^4} + \frac{1}{x^4}}\]Simplifying each term, we get: \[\frac{\frac{2}{x^2} - \frac{3}{x^3} + \frac{5}{x^4}}{1 - \frac{2}{x^3} + \frac{1}{x^4}}\]

Step 3: Evaluate Each Term As \( x \to \infty \)

As \( x \to \infty \), terms like \( \frac{1}{x^2} \), \( \frac{1}{x^3} \), and \( \frac{1}{x^4} \) all approach zero. Therefore, the expression simplifies as follows:\[\lim_{x \to \infty} \frac{0 - 0 + 0}{1 - 0 + 0} = 0\]

Step 4: State the Result

Since all terms in the numerator approach zero, and the denominator approaches 1, the entire expression approaches zero as \( x \to \infty \). The limit is therefore zero.

Key Concepts

Leading Terms in Rational Functions

When evaluating limits at infinity for rational functions, identifying the leading terms is crucial. Leading terms are the terms with the highest exponents on the variable, often the most significant in determining the behavior of the function as the variable approaches infinity. In the example provided, the leading term in the numerator is \(2x^2\), while the leading term in the denominator is \(x^4\).

• The leading terms give us insight into how the degrees of the polynomials compare, which is vital for simplification and evaluating the limits.
• By focusing on these terms, we can ignore smaller terms that contribute less significantly to the function's growth or decay.
• This understanding allows us to determine how the fraction behaves when \(x\) becomes very large.

This step doesn't just simplify computations but also strengthens your intuition about the function's long-term behavior.

Simplifying Expressions by Dividing by Highest Powers

Once leading terms are identified, the next step is simplifying the rational function. Simplification involves dividing every term in both the numerator and denominator by the highest power of \(x\) present in the denominator, which in this case is \(x^4\). This simplification process changes our original problem to:

• \(\frac{\frac{2}{x^2} - \frac{3}{x^3} + \frac{5}{x^4}}{1 - \frac{2}{x^3} + \frac{1}{x^4}}\)
• Here, each term is now expressed with \(x\) in the denominator, and as \(x\) grows larger, these fractional expressions tend toward zero.
• This reveals the true nature of the limit of the function, as the dominant terms are diminished to 0.

Thus, what remains significant in the denominator is \(1\), making it clear how the numerator approaches zero affects the outcome of the limit.

Evaluating Limits at Infinity

After simplifying, the next step is to evaluate each term as \(x\) approaches infinity. For the terms \(\frac{2}{x^2}\), \(\frac{3}{x^3}\), and \(\frac{5}{x^4}\) in the numerator, and \(\frac{2}{x^3}\) and \(\frac{1}{x^4}\) in the denominator, we observe:

• As \(x\) grows, these terms approach zero because dividing a constant by an increasingly large number results in values approaching zero.
• This leaves us with the simplified limit \(\frac{0 - 0 + 0}{1 - 0 + 0}\), which simplifies to \(\frac{0}{1}\), clearly 0.
• This shows that the limit of the whole expression as \(x\) tends towards infinity is zero.

Understanding this calculation confirms that the rational function decreases to zero, showing the value of analytical skills in predicting the behavior of functions at extreme values of \(x\).