Question

Other techniques Evaluate the following limits, where a and \(b\) are fixed real numbers. $$\lim _{x \rightarrow-b} \frac{(x+b)^{7}+(x+b)^{10}}{4(x+b)}$$

Short answer

Question: Find the limit of the function $$\lim _{x \rightarrow-b} \frac{(x+b)^{7}+(x+b)^{10}}{4(x+b)}$$ Answer: The limit of the function as x approaches -b is 0.

Step-by-step solution

Factor out the common factor (x+b)

We can factor out the common factor (x+b) in the numerator: $$\frac{(x+b)^{7}+(x+b)^{10}}{4(x+b)} = \frac{(x+b)^7(1+(x+b)^3)}{4(x+b)}$$

Cancel out the common factors

Now we can cancel out the common factors (x+b) from the numerator and the denominator: $$\frac{(x+b)^7(1+(x+b)^3)}{4(x+b)} = \frac{(x+b)^6(1+(x+b)^3)}{4}$$

Substitute the limit into the simplified expression

As x approaches -b, the expression becomes: $$\lim_{x \rightarrow -b} \frac{(x+b)^6(1+(x+b)^3)}{4} = \frac{(-b+b)^6(1+(-b+b)^3)}{4} = \frac{0}{4}$$

Evaluate the limit

Since the simplified expression is 0, the limit is also 0: $$\lim _{x \rightarrow-b} \frac{(x+b)^{7}+(x+b)^{10}}{4(x+b)} = 0$$

Key concepts

Limit Evaluation

In calculus, evaluating limits is a fundamental concept that helps us understand the behavior of functions as they approach specific points. When we evaluate a limit, we look at what happens to the expression as the variable gets closer and closer to a particular value. For instance, in the exercise we are considering, the limit evaluated is as \( x \) approaches \( -b \). This sort of evaluation is crucial because it allows us to determine if the function converges to a specific value or behaves in another predictable manner.

To effectively evaluate limits, one typically follows specific methods:

• Direct substitution: This method is simplest but only works if the function is continuous and defined at the point of interest.
• Simplifying the expression: Factoring, expanding, or combining like terms can make the limit easier to evaluate, as seen in our original exercise.
• L'Hôpital's Rule: Sometimes helpful for limits resulting in indeterminate forms like \( \frac{0}{0} \), though not used in this particular solution.

By following these steps, we gain insights into the behavior of the functions near the point that is being approached, leading to a better understanding of their overall nature.

Common Factor

A common factor in the context of this calculus problem refers to a shared term that appears in both the numerator and the denominator of a fraction. Identifying and factoring out these common terms is particularly useful in simplifying expressions, as it allows us to cancel them out, leading to a simpler form of the function.

In the given exercise, the common factor is \( (x+b) \). This shared term appears in every term in the numerator as well as in the denominator. By factoring \( (x+b) \) out of the numerator, we reduce the degree of the polynomial, thus simplifying our expression. The formula here initially: \[ \frac{(x+b)^{7}+(x+b)^{10}}{4(x+b)} \]is transformed by factoring out, yielding:\[ \frac{(x+b)^7(1+(x+b)^3)}{4(x+b)} \]

This simplification not only aids in evaluating limits but also generally makes the mathematical handling of problems easier, by reducing the complexity of the terms involved. Recognizing and canceling these common factors is a key step in finding limits for functions represented as fractions.

Limit Simplification

Limit simplification is an essential step in solving limit problems. It involves rewriting the expression to eliminate indeterminate forms, such as \( \frac{0}{0} \), so that the limit can be easily computed. Simplification often makes the expression more manageable and reveals the true behavior of the function near the point of inquiry.

After factoring out common factors, as done in the previous steps, the remaining expression \( \frac{(x+b)^6(1+(x+b)^3)}{4} \) is much simpler and can be easily evaluated by direct substitution. With \( x \) approaching \( -b \), we substitute to find that both \( (x+b)^6 \) and \( (1+(x+b)^3) \) evaluate to 0 as \( x \rightarrow -b \).

This simplification leads to a limit of:\[ \frac{0 \times 1}{4} = 0 \]

Thus, through simplification, we see clearly how the function behaves and we arrive at the conclusion that the limit is \( 0 \). This process underscores the importance of simplification in making complex calculus problems less daunting and more straightforward to solve.