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Chapter 1: Problem 27

Use algebra to find the limit exactly. $$\lim _{x \rightarrow 1} \frac{x^{2}+4}{x+8}$$

Short Answer

Expert verified
The limit is \(\frac{5}{9}\).

Step by step solution

01

Substitute the Limit Value

First, substitute the value of \(x\) that \(x\) approaches, which is 1, into the function to determine if direct substitution can be used. Calculate \(\frac{1^2+4}{1+8}\).
02

Simplify the Expression

Calculate the numerator and the denominator separately. For the numerator \(1^2 + 4 = 5\) and the denominator \(1 + 8 = 9\).
03

Evaluate the Limit

Since the substitution does not result in an indeterminate form, evaluate the expression: \(\frac{5}{9}\).

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Key Concepts

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

Direct Substitution
When evaluating limits, the simplest method to try first is direct substitution. This involves substituting the limit value directly into the function. It's like testing the waters to see if everything works as it should. Suppose you have a function \(f(x)\) and you need to find \(\lim _{x \rightarrow a} f(x)\). Direct substitution means replacing \(x\) with \(a\) in \(f(x)\). If this gives you a real number, then you've found your limit!
  • If the function is continuous at that point, direct substitution will work smoothly.
  • Avoid using this when the substitution results in undefined expressions like \(\frac{0}{0}\).
In our exercise, we substituted \(x = 1\) directly into the expression, and it worked seamlessly without any complications.
Algebra
Algebra plays a crucial role in handling expressions within limits. It often helps in rewriting and simplifying expressions to make them easier to evaluate. This includes standard operations like addition, subtraction, multiplication, and division, as well as applying formulas or expanding expressions. Use algebra to:
  • Simplify expressions to a form where direct substitution is apparent.
  • Factorize or break down complex expressions, especially when they result in indeterminate forms.
  • Combine like terms for easier calculation.
In our example, algebra was used after recognizing that direct substitution could proceed smoothly, without additional manipulation of the given function.
Fraction Simplification
Fraction simplification is the process of reducing a fraction to its simplest form. This is essential in limit problems to make sure you're not accidentally seeing something more complicated than it is.Steps for Simplifying Fractions:
  • Check if the numerator and denominator share any common factors.
  • Divide the numerator and the denominator by their greatest common divisor (GCD), if applicable.
In the exercise provided, once the direct substitution was made, \(\frac{5}{9}\) was the simplest form and required no further simplification. Always remember: if fractions can be reduced further, it might give clearer insights into the behavior of functions as they approach a limit.

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Most popular questions from this chapter

Explain what is wrong with the statement. $$\lim _{x \rightarrow 1} \frac{x-1}{|x-1|}=1$$

For the given \(m\) and \(n\), evaluate $$\lim _{x \rightarrow 1} f(x)$$ or explain why it does not exist, where $$f(x)=\frac{(x-1)^{n}}{(x-1)^{m}}$$ \(n\) and \(m\) are positive integers with \(m=n\)

For the given constant \(c\) and function \(f(x),\) find a function \(g(x)\) that has a hole in its graph at \(x=c\) but \(f(x)=g(x)\) everywhere else that \(f(x)\) is defined. Give the coordinates of the hole. $$f(x)=\sin x, c=\pi$$

Explain what is wrong with the statement. $$\frac{1}{\ln x}=e^{x}$$

Are the statements true or false? Give an explanation for your answer. The function \(f(t)=\sin (0.05 \pi t)\) has period 0.05
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