Problem 22 - A Graphical Approach to Precalculus with Limits

Question

Determine each limit, if it exists. $$\lim _{x \rightarrow 0} \frac{x^{2}+2 x}{x}$$

Step-by-Step Solution

Answer

The limit is 2.

Step 1: Simplify the expression

The given expression is \( \frac{x^2 + 2x}{x} \). We can simplify this expression by dividing the numerator by \( x \), assuming \( x eq 0 \). This gives \( \frac{x(x + 2)}{x} \). The \( x \) in the numerator and denominator cancel out, resulting in \( x + 2 \).

Step 2: Evaluate the limit of the simplified expression

Now, substitute \( x = 0 \) into the simplified expression obtained in Step 1, which is \( x + 2 \). By substituting \( x = 0 \), we get \( 0 + 2 = 2 \).

Step 3: Confirm the existence of the limit

Since we were able to substitute \( x = 0 \) directly after simplifying and found a finite value, the limit exists. The limit is independent of the direction from which \( x \to 0 \) is approached, confirming the existence of the limit.

Key Concepts

Simplifying Expressions

Simplifying expressions is a crucial step when working with limits, especially when you need to handle a function that seems more complicated than it needs to be. In this exercise, the original expression was \( \frac{x^2 + 2x}{x} \). The key to simplifying it was recognizing the common factor of \( x \) in the numerator.

• Factor the numerator: Start by rewriting \( x^2 + 2x \) as \( x(x + 2) \).
• Cancel common factors: By dividing both the numerator and the denominator by \( x \), you cancel out the \( x \), given that \( x eq 0 \). This leads you to the simplified form: \( x + 2 \).

Simplification is essential because it often makes evaluating the limit easier. Remember, the domain of the function may exclude the point you are interested in (here \( x eq 0 \)), but simplification can remove complexities to help you see the solution clearly.

Evaluating Limits

Once you've simplified the expression, evaluating the limit becomes much more straightforward. In the given problem, after simplification, the expression was reduced to \( x + 2 \). Evaluating the limit involves substituting the value that \( x \) is approaching, and observing the behavior of the expression.

• Substitute directly: If the simplified expression isn’t undefined or indeterminate when you plug in the limit, you can substitute directly. For this problem, substitute \( x = 0 \) into \( x + 2 \).
• Calculate: The substitution yields \( 0 + 2 \), resulting in \( 2 \).

Direct substitution works perfectly here because the expression is continuous and doesn’t create an undefined situation at \( x = 0 \). This makes evaluation simple and demonstrates the power of prior simplification.

Existence of Limits

It’s not enough to simply calculate a value for a limit; you must also examine whether the limit exists. A limit exists if the function approaches a specific value from both directions, as \( x \) approaches the point of interest. Here, after simplifying, you get \( x + 2 \), which is a continuous function at \( x = 0 \).

• Continuity check: Since \( x + 2 \) doesn’t have any discontinuities at \( x = 0 \), the substitution confirms the function approaches \( 2 \) from both directions.
• Direction independence: The limit consistent regardless of approaching from the left or the right of \( x = 0 \), further confirming the limit’s existence.

In conclusion, establishing the existence of a limit ensures that the solution isn’t just numerically correct, but also mathematically sound, confirming \( \lim_{x \to 0} (x + 2) = 2 \) holds true.