Question

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

Answer: The limit of the given function as x approaches -b is 0.

Step-by-step solution

Factor out (x+b) from both the numerator and the denominator

We factor out (x+b) from each term in the numerator and the denominator to simplify the expression as follows: $$ \frac{(x+b)^7 + (x+b)^{10}}{4(x+b)} = \frac{(x+b)((x+b)^6 + (x+b)^9)}{4(x+b)} $$

Cancel out the common term

Notice that both the numerator and the denominator have a common term, (x+b). We can cancel out this term from both the numerator and the denominator: $$ \frac{(x+b)((x+b)^6 + (x+b)^9)}{4(x+b)} = \frac{(x+b)^6 + (x+b)^9}{4} $$

Substitute the limit value

Now, we can substitute the limit value x=-b in the simplified expression: $$ \lim_{x\rightarrow -b} \frac{(x+b)^6 + (x+b)^9}{4}=\frac{(-b+b)^6+(-b+b)^9}{4} $$

Simplify the expression

Since both terms in the numerator will become 0, the expression simplifies to: $$ \frac{0^6 + 0^9}{4} = \frac{0 + 0}{4} = 0 $$

Final answer

The limit of the given function as x approaches -b is: $$ \lim_{x\rightarrow -b} \frac{(x+b)^7+(x+b)^{10}}{4(x+b)}=0 $$

Key concepts

Algebraic Simplification

Algebraic simplification is a crucial step in evaluating complex expressions. It involves breaking down an expression into its simplest form, making it easier to understand and solve. When dealing with limits, as in the given exercise, simplifying the algebraic expression can avoid direct substitution that leads to undefined forms such as \(\frac{0}{0}\).

To simplify effectively, identify common factors in the numerator and the denominator. In our problem, \((x + b)\) was a common factor. By factoring it out, we reduced a complex polynomial expression to something more manageable. Once simplified, substitution becomes straightforward, avoiding unnecessary complications. Simplifying algebraic expressions not only makes calculations easier but also prevents errors that may arise from handling unsimplified forms.

Limits of Functions

Limits help us understand the behavior of functions as the input approaches a certain value. They are fundamental to calculus and essential for understanding continuity and changes at specific points.

In this exercise, we are tasked with finding \(\lim_{x \to -b} \frac{(x+b)^7 + (x+b)^{10}}{4(x+b)}\). The limit focuses on how the expression behaves as \( x \) approaches \(-b\). Through simplification, we reach an expression without undefined terms at \( x = -b \). This allows us to directly substitute \( x = -b \) to find the limit.

Understanding the concept of limits is vital in higher mathematics. It provides insights into the function's continuity and its value at points inaccessible through direct evaluation due to division by zero or other undefined states. Limits serve as a bridge between discrete points and the continuous nature of calculus.

Factoring Expressions

Factoring involves rewriting an expression as a product of its factors, making it easier to manage and simplify. This skill is essential in solving equations and inequalities, particularly when working with polynomials.

In our exercise, factoring was used to simplify the expression by removing the common factor \((x+b)\). By identifying and factoring out \((x+b)\), we reduced the expression significantly and canceled out terms that were obstructing our path to finding the limit.

• Familiarize with various factoring techniques, such as GCF, difference of squares, and trinomial factoring.
• Always check for common factors in both the numerator and denominator when simplifying expressions involving fractions.
• Factoring simplifies expressions by reducing complexity, making other operations like finding limits or solving equations more straightforward.

Factoring is not just a cleanup tool; it's a powerful method in algebra for transforming complex expressions into simpler ones, which is particularly useful in calculus and further mathematical studies.