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Chapter 10: Problem 48

Calculate the limits in Exercises 21-72 algebraically. If a limit does not exist, say why. $$ \lim _{x \rightarrow-1} \frac{x^{2}+3 x}{x^{2}+3 x+2} $$

Short Answer

Expert verified
The limit does not exist because when substituting x = -1 into the function \(f(x) = \frac{x(x + 3)}{(x + 1)(x + 2)}\), it causes the denominator to become zero, resulting in an undefined value. In general, a limit does not exist if the function leads to a division by zero or if the function becomes infinitely large or infinitely small near the given point. In this case, our function leads to a division by zero.

Step by step solution

01

Identify the function and the given limit

We are given a rational function: $$ f(x) = \frac{x^2 + 3x}{x^2 +3x + 2} $$ We are asked to find the limit as x approaches -1: $$ \lim_{x \rightarrow -1} f(x) $$
02

Factor the numerator and the denominator

Before we substitute the value of x to find the limit, let's factor the numerator and the denominator. Numerator: $$ x^2 + 3x = x(x + 3) $$ Denominator: $$ x^2 + 3x + 2 = (x + 1)(x + 2) $$ Now rewrite the original function using the factored forms: $$ f(x) = \frac{x(x + 3)}{(x + 1)(x + 2)} $$
03

Check for common factors

We see that there are no common factors between the numerator and the denominator. Thus, the function stays as: $$ f(x) = \frac{x(x + 3)}{(x + 1)(x + 2)} $$
04

Calculate the limit

Now, we can substitute the value of x (-1) into the function to find the limit: $$ \lim_{x \rightarrow -1} f(x) = \frac{-1(-1 + 3)}{(-1 + 1)(-1 + 2)} $$ As we can see, when substituting x = -1, we get division by zero in the denominator. Therefore, the limit does not exist.
05

Explain why the limit does not exist

The limit does not exist because when substituting x = -1 into the function, it causes the denominator to become zero, resulting in an undefined value. In general, a limit does not exist if the function leads to a division by zero or if the function becomes infinitely large or infinitely small near the given point. In this case, our function leads to a division by zero.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rational Functions
Rational functions are a key topic in algebra and consist of a ratio of two polynomials. In simpler terms, a rational function is essentially a fraction where both the numerator and the denominator are polynomials. These functions are written in the form \( f(x) = \frac{P(x)}{Q(x)}\).
Understanding rational functions is important because they often come up in calculus, algebra, and various real-world situations.
  • The function can be evaluated at specific values to determine points on its graph.
  • Analyzing rational functions helps in understanding their behavior, such as asymptotes and discontinuities.
When calculating limits for rational functions, what we're essentially doing is investigating the function's behavior as the variable (\(x\)) approaches a particular value. This exploration often reveals more about the function's tendencies.
Factoring Polynomials
Factoring polynomials is an essential skill when dealing with algebraic expressions and specifically rational functions. It involves breaking down a polynomial into simpler terms, or 'factors,' that can still be multiplied together to return the original polynomial. This process is fundamental in simplifying expressions and solving polynomial equations.
In the example given:
  • The numerator \(x^2 + 3x\) factors into \(x(x + 3)\).
  • The denominator \(x^2 + 3x + 2\) factors into \((x + 1)(x + 2)\).
By factoring, we can simplify complex expressions, which is especially useful when assessing limits in rational functions. Factorization reveals common factors that could be canceled, but when none exist鈥攁s in this case鈥攊t tells us more about the behavior of the function's limit as well as potential points of discontinuity.
Division by Zero
Division by zero is a critical topic in mathematics and one that can lead to undefined expressions, making it unique compared to other operations. When a function involves division by zero, it means a value results in an infinite or undefined outcome, which typically suggests a discontinuity at that point.
In our initial problem:
  • When \(x = -1\) is substituted, the denominator becomes \(0\), leading to division by zero.
  • This causes the expression to be undefined, meaning the limit does not exist.
The concepts of limits often involve managing situations where denominators become zero. A function may approach infinity (or negative infinity) near such points, indicating an asymptote on the graph. Understanding these behaviors is crucial for developing a richer insight into functions and helping navigate the complexities of calculus.

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Most popular questions from this chapter

(Compare Exercise 30 of Section 10.4.) The value of subprime (normally classified as risky) mortgage debt outstanding in the U.S. can be approximated by $$ A(t)=\frac{1,350}{1+4.2(1.7)^{-t}} \text { billion dollars } \quad(0 \leq t \leq 9) $$ where \(t\) is the number of years since the start of 2000 . a. Estimate \(A(7)\) and \(A^{\prime}(7)\). (Round answers to three significant digits.) What do the answers tell you about subprime mortgages? b. I Graph the function and its derivative and use your graphs to estimate when, to the nearest year, \(A^{\prime}(t)\) is greatest. What does this tell you about subprime mortgages? HINT [See Example 5.]

Compute the derivative function \(f^{\prime}(x)\) algebraically. (Notice that the functions are the same as those in Exercises \(1-14 .)\) HINT [See Examples 2 and \(3 .]\) $$ f(x)=3 x-4 $$

Compute the derivative function \(f^{\prime}(x)\) algebraically. (Notice that the functions are the same as those in Exercises \(1-14 .)\) HINT [See Examples 2 and \(3 .]\) $$ f(x)=-2 x+4 $$

In Exercises 73 and 74, use technology to graph the derivative of the given function for the given range of values of \(x .\) Then use your graph to estimate all values of \(x\) (if any) where the tangent line to the graph of the given function is horizontal. Round answers to one decimal place. HINT [See Example 4.] $$ f(x)=x^{4}+2 x^{3}-1 ; \quad-2 \leq x \leq 1 $$

Use the method of Example 4 to list approximate values of \(f^{\prime}(x)\) for \(x\) in the given range. Graph \(f(x)\) together with \(f^{\prime}(x)\) for \(x\) in the given range. $$ f(x)=\frac{10 x}{x-2} ; \quad 2.5 \leq x \leq 3 $$
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