Problem 20 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

Either find the given limit or show it does not exist. If the limit is infinite, indicate whether it is \(+\infty\) or \(-\infty\). $$ \lim _{x \rightarrow 2} \frac{x^{2}-3 x}{x+1} $$

Step-by-Step Solution

Answer

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

Step 1: Substitute the limit

Substitute the value of x approaching 2 directly into the expression \( \frac{x^2 - 3x}{x + 1} \). This gives \( \frac{2^2 - 3 \cdot 2}{2 + 1} = \frac{4 - 6}{3} = \frac{-2}{3} \).

Step 2: Simplify the function

Recognize that there are no indeterminate forms and simplifying the numerator gives \( x^2 - 3x = x(x - 3) \), but direct substitution is sufficient here.

Step 3: Confirm the limit

Because direct substitution yields the result \( \frac{-2}{3} \), we conclude that the limit exists and is \( \frac{-2}{3} \).

Key Concepts

limit of a function

In calculus, one of the fundamental concepts is the 'limit of a function'. When we talk about the limit, we mean the value that a function approaches as the input approaches some value.
Limits are essential because they help us understand the behavior of functions, especially those functions that do not have a clear output at certain points.
Formally, the limit of a function f(x) as x approaches a value 'a' is denoted as: \ \( \lim_ {x \rightarrow a} f(x) = L \ \)
Here, 'L' is the value that f(x) approaches as x gets closer and closer to 'a'.

direct substitution method

One of the simplest methods to determine the limit of a function is through the 'direct substitution method'. This involves substituting the value of the variable directly into the function.
This method works best when the given function is continuous at the point being evaluated.
For example, consider the given exercise: \ \( \lim _ {x \rightarrow 2} \frac{ x^2 - 3x } { x + 1 } \ \).
By directly substituting x = 2 into the expression, we get:
\ \( \frac{ 2^2 - 3 \ \cdot 2 } { 2 + 1 } = \frac{ 4 - 6 } { 3 } = \frac{ -2 } { 3 } \ \).
Since there are no indeterminate forms in this case, direct substitution gives us the limit directly.

determine limit in calculus

To determine the limit in calculus, follow these steps:

• Identify the point at which you need to find the limit.
• Substitute that point into the function to see if it produces a valid output.
• If substitution leads to an indeterminate form (like 0/0), you may need to simplify the function or use other techniques like factoring, rationalizing, or applying L'Hôpital's rule.

For our exercise:

• We wanted to find the limit as x approaches 2.
• By directly substituting x= 2, the function simplifies smoothly, without needing further techniques.
• Thus, our determined limit is \ \( \frac { -2 } { 3 } \ \).