Problem 14 - Calculus Early Transcendentals

Question

Evaluate the following limits using \(l\) Hópital's Rule. $$\lim _{x \rightarrow-1} \frac{x^{4}+x^{3}+2 x+2}{x+1}$$

Step-by-Step Solution

Answer

Answer: The limit of the function as x approaches -1 is 1. Explanation: By recognizing the indeterminate form and applying L'Hôpital's Rule, we find that the limit of the given function as x approaches -1 is equal to 1.

Step 1: Recognize the indeterminate form

As x approaches -1, the given function becomes: $$\frac{(-1)^4 + (-1)^3 + 2(-1) + 2}{(-1) + 1} =\frac{0}{0}$$ This is an indeterminate form of 0/0, so we can apply L'Hôpital's Rule.

Step 2: Take the derivatives of the numerator and the denominator

We take the derivative of the numerator and the denominator with respect to x: $$\frac{d}{dx}(x^4 + x^3 + 2x + 2) = 4x^3 + 3x^2 + 2$$ $$\frac{d}{dx}(x+1) = 1$$

Step 3: Apply L'Hôpital's Rule

Now, we apply L'Hôpital's Rule and evaluate the limit of the new quotient as x approaches -1: $$\lim_{x\to-1} \frac{4x^3 + 3x^2 + 2}{1}$$

Step 4: Evaluate the limit

The function is now continuous at x = -1, so we substitute x = -1 directly into the expression: $$4(-1)^3 + 3(-1)^2 + 2 = -4 + 3 + 2 = 1$$ Therefore, the limit of the given function as x approaches -1 is 1: $$\lim _{x \rightarrow-1} \frac{x^{4}+x^{3}+2 x+2}{x+1} = 1$$

Key Concepts

Indeterminate Forms

When we talk about indeterminate forms in calculus, we are referring to specific expressions that do not have a clear or defined limit. These forms often arise when evaluating the limits of functions and can include expressions like \(\frac{0}{0}\), \(\frac{\infty}{\infty}\), \(0\cdot\infty\), \(\infty - \infty\), \(0^0\), \(\infty^0\), and \(1^\infty\).

When you encounter an indeterminate form while calculating a limit, it signals that more work needs to be done to find out what the function is actually approaching. You cannot conclude the behaviour of the function from the indeterminate form alone; it requires further analysis, often through algebraic manipulation or the application of specific limit theorems such as L'Hôpital's Rule. In the example of \(\frac{x^{4}+x^{3}+2x+2}{x+1}\) as \({x \rightarrow-1}\), the direct substitution results in \(\frac{0}{0}\), which is an indeterminate form that L'Hôpital's Rule can help resolve.

Calculus Limits

A limit in calculus is a fundamental concept that describes the value that a function approaches as the input (or 'x' value) approaches a certain point. Limits are essential for understanding the behavior of functions, particularly near points that are not clearly defined or at the boundaries of the function's domain.

Limits deal with the expected result as a point gets arbitrarily close to, but not necessarily reaches, a certain value. They allow mathematicians to rigorously define and work with concepts like continuity, derivatives, and integrals. Calculating limits can often be straightforward through direct substitution. However, when this direct approach leads to indeterminate forms, we must apply techniques such as factoring, rationalizing, or using L'Hôpital's Rule, a process that leverages the relationship between limits and derivatives to find the desired limit. For example, L'Hôpital's Rule is adept at resolving limits that initially produce the indeterminate form \(\frac{0}{0}\) upon direct substitution.

Derivative

The derivative of a function represents the rate at which the function's value changes with respect to a change in its input value. Geometrically, the derivative is the slope of the tangent line to the curve of the function at a particular point. In a practical sense, if you have a function that denotes the position of a car over time, the derivative of that function would give you the car's velocity.

In the context of solving indeterminate limit problems, derivatives are especially important. When applied as part of L'Hôpital's Rule, taking the derivative of both the numerator and the denominator can often simplify the evaluation of a limit that is otherwise indeterminate. The process involves differentiating until the limit can be computed through direct substitution or until the indeterminate form is eliminated. Continuously applying this rule and taking derivatives, as done in our example function \(\frac{x^{4}+x^{3}+2x+2}{x+1}\), streamlines reaching the solution and calculating the precise limit, thereby providing us with the value that the function is approaching.