Problem 16 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the indicated limit or state that it does not exist. In many cases, you will want to do some algebra before trying to evaluate the limit. $$\lim _{x \rightarrow-1} \frac{x^{2}+x}{x^{2}+1}$$

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Identify Direct Substitution

First, we attempt to evaluate the limit using direct substitution by substituting \(x = -1\) into the function. This means we calculate \(\frac{(-1)^{2} + (-1)}{(-1)^{2} + 1}\).

Step 2: Simplify the Expression

Simplify the expression obtained from direct substitution: \((-1)^{2} + (-1) = 1 - 1 = 0\) and \((-1)^{2} + 1 = 1 + 1 = 2\). Thus, the expression becomes \(\frac{0}{2}\).

Step 3: Evaluate the Limit

Since the expression simplified to \(\frac{0}{2}\), the result is \(0\). This can be directly evaluated as an integer result for the given limit.

Key Concepts

Direct Substitution

Direct Substitution is a common method for evaluating limits in calculus. It involves directly substituting the limit point into the function where the variable is replaced with a specific number. This technique is often the first step when trying to determine the value of a limit.

The process is straightforward:

• Identify the point to which the variable is approaching (e.g., in our problem, this is where \(x\) approaches \(-1\)).
• Substitute this point directly into the function.
• Simplify the result, if possible.

If the expression gives a well-defined number, then the limit equals that value. If it results in an indeterminate form like \(\frac{0}{0}\), further steps like algebraic simplification or L'Hôpital's Rule may be needed. In our example, substituting \(x = -1\) leads directly to evaluating the simplified value \(\frac{0}{2}\), equalling \(0\). This demonstrates that, when applicable, direct substitution offers a quick solution to finding the limit.

Simplifying Rational Expressions

Simplifying Rational Expressions often plays a crucial role when evaluating limits, especially if direct substitution leads to an indeterminate form. A rational expression is a fraction where both the numerator and the denominator are polynomials.

To simplify these expressions:

• Perform algebraic manipulations, such as factoring polynomials, cancelling common factors, and simplifying terms.
• Transform the expression into an equivalent form that doesn't lead to indeterminate results when substituting limit points.

In the problem you're working on, the expression to evaluate is \(\frac{x^2 + x}{x^2 + 1}\). After using direct substitution, it simplifies directly to \(\frac{0}{2}\). However, had this resulted in an undefined form like \(\frac{0}{0}\), further algebraic manipulation such as factorization might have been required to remove such indeterminate forms. Simplifying rational expressions can thus transform complex limits into solvable forms.

Calculus Problem Solving

Calculus Problem Solving involves a structured approach to tackle problems posed in calculus, especially when dealing with limits. The key is to strategize your solution depending on the nature of the problem. For evaluating limits:

• Firstly, attempt Direct Substitution to see if a specific limit value can be quickly obtained.
• If Direct Substitution leads to indeterminate forms, explore other strategies such as simplification, factoring, and applying L'Hôpital's Rule.
• Consistently review your calculations to ensure each step logically follows from the previous.

Understanding the foundational aspects of calculus like limits helps solve other discussions on continuity, derivatives, and integrals. Approaching each problem with this mindset makes solving calculus exercises manageable. In our scenario, after simplifying the expression using direct substitution in combination with algebraic techniques, we efficiently reach the solution of \(0\). This demonstrates how calculus problem-solving skills guide steps toward the correct solution.