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Chapter 2: Problem 7

Suppose \(p\) and \(q\) are polynomials. If \(\lim _{x \rightarrow 0} \frac{p(x)}{q(x)}=10\) and \(q(0)=2,\) find \(p(0)\).

Short Answer

Expert verified
Answer: $p(0) = 20$.

Step by step solution

01

Apply the limit definition

According to the problem, we have \(\lim_{x \rightarrow 0} \frac{p(x)}{q(x)} = 10\). Since the limit is finite and non-zero, we can apply the limit definition and break it down as follows: \(\lim_{x \rightarrow 0} \frac{p(x)}{q(x)} = \frac{\lim_{x \rightarrow 0} p(x)}{\lim_{x \rightarrow 0} q(x)} = 10\)
02

Substitute the given information

We are given that \(q(0) = 2\). Thus, we can substitute this value in the limit formula: \(\frac{\lim_{x \rightarrow 0} p(x)}{2} = 10\) Now, we know that the value of \(\lim_{x \rightarrow 0} p(x)\) is nothing but \(p(0)\). So, we substitute this value in the formula:
03

Solve for \(p(0)\)

Now that we have the appropriate equation, we solve for \(p(0)\): \(\frac{p(0)}{2} = 10\) Multiply both sides by 2: \(p(0) = 20\) Hence, we have determined that \(p(0) = 20\).

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Key Concepts

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

Polynomial Division
Polynomial division is a mathematical process used to divide one polynomial by another. It is similar to long division with numbers. You break down a polynomial into simpler components via a method known as polynomial division or synthetic division.
  • Polynomials are expressions consisting of variables and coefficients combined using addition, subtraction, multiplication, and non-negative integer exponents.
  • Common polynomial division involves dividing a polynomial dividend by a polynomial divisor.
  • This method helps simplify expressions and solve polynomial equations.
When we encounter problems where we need to divide two polynomials, it can help simplify expressions before applying other mathematical techniques like limits. Understanding division aids in analyzing and solving calculus-based problems involving polynomials.
Limits and Continuity
Limits are fundamental in calculus and help us understand how functions behave as they approach a specific point. The idea of limits is crucial in defining derivatives and integrals.
  • A limit describes the value that a function approaches as the input approaches some point.
  • Limits are essential for determining the behavior of functions at points where they might not be directly defined.
  • Continuity means a function is smooth without breaks, jumps, or holes in its graph.
In solving problems like finding the limit of the expression \(\lim_{x \to 0} \frac{p(x)}{q(x)} = 10\), knowing that the limit exists and is finite helps us conclude the value the function approaches as \(x\) approaches a certain number. This requires understanding both the numerator and the denominator behavior at that point.
Problem Solving in Calculus
Problem-solving in calculus involves applying different mathematical concepts and techniques to find solutions. It requires breaking down complex problems into simpler parts.
  • Identifying given information and what needs to be found is crucial in setting a clear path to a solution.
  • Using limit properties aids in simplifying expressions and evaluating them at specific points.
  • Effective problem-solving often requires leveraging calculus rules like the limit of a quotient which is the quotient of the limits.
In the example problem, we observe how properties of limits and continuity guide us to the solution. We learn to isolate and identify values using initial conditions, like \(q(0) = 2\), to eventually solve for \(p(0)\). Such exercises sharpen analytic skills and prepare students to tackle more complex calculus problems.

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Most popular questions from this chapter

Theorem 4a Given the polynomial $$p(x)=b_{n} x^{n}+b_{n-1} x^{n-1}+\cdots+b_{1} x+b_{0}$$ prove that \(\lim _{x \rightarrow a} p(x)=p(a)\) for any value of \(a\)

Asymptotes Use analytical methods and/or a graphing utility to identify the vertical asymptotes (if any) of the following functions. $$g(\theta)=\tan \left(\frac{\pi \theta}{10}\right)$$

Asymptotes Use analytical methods and/or a graphing utility to identify the vertical asymptotes (if any) of the following functions. $$f(x)=\frac{1}{\sqrt{x} \sec x}$$

Analyzing infinite limits graphically Graph the function \(y=\tan x\) with the window \([-\pi, \pi] \times[-10,10] .\) Use the graph to analyze the following limits. a. \(\lim _{x \rightarrow \pi / 2^{+}} \tan x\) b. \(\lim _{x \rightarrow \pi / 2^{-}} \tan x\) d. \(\quad \lim _{n}\) tan \(x\) c. \(\lim _{x \rightarrow-\pi / 2^{+}} \tan x\) d. \(\lim _{x \rightarrow-\pi / 2^{-}} \tan x\)

Determine the end behavior of the following transcendental functions by evaluating appropriate limits. Then provide a simple sketch of the associated graph, showing asymptotes if they exist. $$f(x)=\frac{50}{e^{2 x}}$$
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