Problem 2 - Calculus with Concepts in Calculus

Question

Use the results of this section to evaluate the limit. $$ \lim _{x \rightarrow \sqrt{2}}\left(x^{2}+5\right)(\sqrt{2} x+1) $$

Step-by-Step Solution

Answer

The limit is 21.

Step 1: Understand the Problem

We need to find the limit of \((x^2 + 5)(\sqrt{2} x + 1)\) as \(x\) approaches \(\sqrt{2}\). This involves evaluating the expression at a value very close to \(\sqrt{2}\).

Step 2: Direct Substitution

Since \(x^2 + 5\) and \(\sqrt{2}x + 1\) are both continuous functions, we can directly substitute \(x = \sqrt{2}\) into the expression. So the limit can be calculated by substituting \(x = \sqrt{2}\) into \((x^2 + 5)(\sqrt{2} x + 1)\).

Step 3: Substitute and Calculate

Substitute \(x = \sqrt{2}\) into \(x^2 + 5\) to get \((\sqrt{2})^2 + 5 = 2 + 5 = 7\). Then substitute \(x = \sqrt{2}\) into \(\sqrt{2}x + 1\) to get \(\sqrt{2} \times \sqrt{2} + 1 = 2 + 1 = 3\).

Step 4: Multiply the Results

Multiply the results obtained from the previous step: \(7 \times 3 = 21\). This gives the value of the limit \(\lim _{x \rightarrow \sqrt{2}} (x^{2} + 5)(\sqrt{2} x + 1) = 21\).

Key Concepts

Direct Substitution Method

The direct substitution method is a straightforward technique used to evaluate limits in calculus. When you encounter continuous functions, this method allows you to substitute the value that \( x \) approaches directly into the function. This is simple and effective when dealing with polynomials, rational, exponential, and trigonometric functions that are continuous at the point of evaluation.

For example, with the expression \((x^2 + 5)(\sqrt{2} x + 1)\) and \( x \) approaching \( \sqrt{2} \), both parts of the expression—\( x^2 + 5 \) and \( \sqrt{2}x + 1 \)—are continuous functions. By using the direct substitution method, you can substitute \( x = \sqrt{2} \) directly into these functions to find the limit. This approach saves time and avoids unnecessary complications, as it circumvents the need for more advanced limit techniques when not required.

Continuity of Functions

Understanding the continuity of functions is crucial for applying the direct substitution method effectively. A function is continuous at a point if there is no interruption in its graph at that point. Simply put, it's like tracing the curve on a graph without lifting your pencil.

Polynomials and functions formed from them, like \( x^2 + 5 \), are examples of continuous functions. For the function \( \sqrt{2} x + 1 \), continuity means you can smoothly plug in \( x = \sqrt{2} \) and expect the function to behave predictably. The values of functions at continuous points truly reflect the behavior of the function around those points.

• No sudden jumps
• No breaks or holes in the graph
• The function is defined and finite at that point

These characteristics ensure that when using the direct substitution method, the calculated limit will be accurate.

Evaluating Limits

Evaluating limits is a fundamental concept in calculus, which involves finding out what value a function approaches as the variable within it gets close to a certain point. This helps in understanding the behavior of functions at and near specific values.

In the exercise, you started by considering the limit \( \lim_{x \to \sqrt{2}}(x^2 + 5)(\sqrt{2}x + 1) \). By breaking it down into products of simpler expressions—both continuous—you used direct substitution to calculate the limit effectively. Here's a breakdown of what happens:

• First, substitute into \( x^2 + 5 \) to find it approaches 7.
• Substitute into \( \sqrt{2}x + 1 \) to find it approaches 3.
• Multiply these results to get the final limit: 21.

Evaluating limits is about understanding function behavior fully, and with continuous functions, it often simplifies into direct substitution, as it did in this exercise.