Problem 9 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

The limit is \( \infty \).

Step 1: Analyze the Limit Expression

We are given the expression \( \lim_{x \rightarrow -\infty} \frac{x^{2} - 3x + 1}{4 - x} \). To evaluate this, we need to understand how the numerator and denominator behave as \( x \) approaches \( -\infty \). The dominating terms in the numerator and denominator are \( x^2 \) and \( -x \), respectively, since they will grow significantly larger in magnitude compared to other terms as \( x \rightarrow -\infty \).

Step 2: Simplify by Dividing Terms

To facilitate evaluating the limit as \( x \rightarrow -\infty \), divide each term in the fraction by \( x \), the variable with the highest degree in the denominator. Thus, the expression becomes: \[ \lim_{x \rightarrow -\infty} \frac{x^{2}/x - 3x/x + 1/x}{4/x - x/x} \].

Step 3: Simplify Further

After dividing each term by \( x \), simplify further:\[ \lim_{x \rightarrow -\infty} \frac{x - 3 + 1/x}{4/x - 1} \].For \( x \rightarrow -\infty \), \( 1/x \) and \( 4/x \) both tend towards zero.

Step 4: Calculate the Limit

With the simplified expression \( \frac{x - 3 + 0}{0 - 1} = \frac{x - 3}{-1} \), which simplifies to \(-(x - 3)\). Therefore, as \( x \rightarrow -\infty \), we now have \(-x + 3 \approx \infty\) because the term \( x \) dominates for sufficiently large magnitudes.

Key Concepts

Evaluating Limits

In calculus, evaluating limits is a fundamental concept that helps us understand the behavior of functions as a variable approaches a certain value. For example, consider the problem \( \lim_{x \rightarrow -\infty} \frac{x^{2} - 3x + 1}{4 - x} \). Evaluating this limit requires examining how both the numerator and the denominator behave as \( x \) tends to \( -\infty \). To accurately find the limit, we need to simplify the expression by focusing on the terms that tend to dominate as \( x \) moves in this direction.

• First, identify how each polynomial term in both the numerator and denominator behaves at extreme \( x \) values.
• Analyze the largest degree terms since these will mainly determine the function's behavior for large \( |x| \).

By systematically breaking down the expression and simplifying it, we can better understand the curve's long-term behavior and evaluate its limit confidently.

Dominating Terms

When dealing with limits involving polynomials, identifying the dominating terms is crucial, especially as \( x \rightarrow \pm \infty \). These are the terms that significantly influence the value of the expression due to their power or degree.For instance, in the expression \( \frac{x^{2} - 3x + 1}{4 - x} \), the dominating term in the numerator is \( x^2 \), and in the denominator, it is \( -x \). These terms rise sharply in magnitude compared to other terms as \( x \) grows large.

• The term \( x^2 \) grows faster than all other terms in the numerator.
• The term \( -x \) in the denominator will also increase significantly in magnitude, affecting the fraction's overall value.

By focusing on the dominating terms, we can often ignore smaller order terms when \( x \) reaches very large values, making it easier to approximate and understand the limit. This simplification helps us predict what happens as \( x \) moves towards \( -\infty \) effectively.

Limit at Infinity

Understanding limits at infinity is essential for predicting the end behavior of functions as they stretch towards extremely large or small values. In the limit \( \lim_{x \rightarrow -\infty} \frac{x^{2} - 3x + 1}{4 - x} \), as \( x \) approaches \( -\infty \), it's vital to focus on which terms will drive the value of the fraction.As we observed earlier, we simplify the rational function by dividing each term by the highest power of \( x \) present in the denominator. This yields:\[ \lim_{x \rightarrow -\infty} \frac{x - 3 + 0}{0 - 1} \]

• The terms \( \frac{1}{x} \) and \( \frac{4}{x} \) approach zero as \( x \rightarrow -\infty \).
• The significant term now in controlling the limit is the \( x \) from the numerator resulting in the expression simplifying to \(-x + 3\).

Ultimately, as \( x \rightarrow -\infty \), because of the term \(-x\), the limit becomes approximately \( \infty \), providing a powerful insight into the behavior of functions in these extreme contexts.