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Chapter 2: Problem 12

Use the definition of continuity and the properties of limits to show that the function is continuous at the given number \(a\). $$ g(t)=\frac{t^{2}+5 t}{2 t+1}, \quad a=2 $$

Short Answer

Expert verified
The function \( g(t) \) is continuous at \( t = 2 \) because \( \lim_{t \to 2} g(t) = \frac{14}{5} = g(2) \).

Step by step solution

01

Understand the Definition of Continuity

A function \( g(t) \) is considered continuous at \( t = a \) if \( \lim_{t \to a} g(t) = g(a) \). This means we need to show that the limit as \( t \) approaches \( a \) equals the value of the function at \( a \).
02

Calculate the Value of the Function at \( t = a \)

Plug \( a = 2 \) into the function \( g(t) \) to find \( g(2) \).\[ g(2) = \frac{2^2 + 5 \cdot 2}{2 \cdot 2 + 1} = \frac{4 + 10}{4 + 1} = \frac{14}{5} \]
03

Calculate the Limit of the Function as \( t \to a \)

Calculate \( \lim_{t \to 2} g(t) \) using the same function.\[ \lim_{t \to 2} \frac{t^2 + 5t}{2t + 1} = \frac{2^2 + 5 \cdot 2}{2 \cdot 2 + 1} = \frac{14}{5} \]Here, direct substitution is valid since the denominator \( 2t + 1 \) does not become zero (\( 2 \times 2 + 1 = 5 eq 0 \)).
04

Compare the Limit with the Function Value

Check if the limit equals the value of the function at \( t = 2 \). Since:\[ \lim_{t \to 2} g(t) = \frac{14}{5} = g(2) \]The limit matches the function value.

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Key Concepts

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

Properties of Limits
Understanding the properties of limits is crucial when evaluating the continuity of functions in calculus. Limits help us understand how functions behave near specific points. Some important properties of limits include:
  • Limit of a Sum: The limit of the sum of two functions is equal to the sum of their limits. Mathematically, \( \lim_{x \to a} (f(x) + g(x)) = \lim_{x \to a} f(x) + \lim_{x \to a} g(x) \).
  • Limit of a Product: The limit of the product of two functions is equal to the product of their limits, i.e., \( \lim_{x \to a} (f(x) \cdot g(x)) = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x) \).
  • Limit of a Quotient: The limit of a quotient is the quotient of their limits, given the function in the denominator does not approach zero, as expressed by \( \lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)} \), if \( \lim_{x \to a} g(x) eq 0 \).
These principles were employed in the solution to evaluate the limit of the function \( g(t) \) as \( t \) approaches 2.
This allowed us to confirm the function's behavior near the point \( a = 2 \).
Definition of Continuity
The concept of continuity in mathematics determines whether a function maintains a smooth and unbroken path over its domain.
A function \( g(t) \) is continuous at a point \( t = a \) if the following three conditions are met:
  • The function \( g(t) \) is defined at \( t = a \). For \( g(t) \) to be continuous at \( t = 2 \), \( g(2) \) must be a defined and real number.
  • The limit \( \lim_{t \to a} g(t) \) exists. We must be able to find the limit of \( g(t) \) as \( t \) approaches 2.
  • The function value and the limit are equal. This means \( \lim_{t \to a} g(t) = g(a) \).
Ensuring these conditions demonstrate that \( g(t) \) is continuous at \( t = 2 \). In our original exercise, steps to check the third condition confirmed that the limit and the value at \( t = 2 \) both equaled \( \frac{14}{5} \). Thus, \( g(t) \) is continuous at that point.
Evaluating Limits
Evaluating limits is a foundational activity in understanding calculus.
This involves finding the function鈥檚 behavior as the input nears a specific point.
There are several techniques for evaluating limits, such as:
  • Direct Substitution: When no undefined form occurs, substitute the target input directly into the function.
  • Factoring: For complex rational functions, factor out both the numerator and the denominator to simplify and eliminate any apparent discontinuities.
  • Conjugate Multiplication: Useful for functions involving squar roots, where multiplying by the conjugate can simplify the limit expression.
  • Limit Laws and Properties: As previously mentioned, these laws can simplify and assist in breaking down functions into manageable parts.
In the worked solution for \( g(t) \), direct substitution was immediately viable since plugging in \( t = 2 \) did not result in a division by zero.
This simplicity illustrated one of the more straightforward methods in evaluating limits.

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The unemployment rate \(U(t)\) varies with time. The table gives the percentage of unemployed in the US labor force from 2003 to 2012 . (a) What is the meaning of \(U^{\prime}(t) ?\) What are its units? (b) Construct a table of estimated values for \(U^{\prime}(t)\). \(\begin{array}{|c|c|c|c|}\hline t & {U(t)} & {t} & {U(t)} \\ \hline 2003 & {6.0} & {2008} & {5.8} \\ {2004} & {5.5} & {2009} & {9.3} \\ {2005} & {5.1} & {2010} & {9.6} \\ {2006} & {4.6} & {2011} & {8.9} \\ {2007} & {4.6} & {2012} & {8.1} \\ \hline\end{array}\)

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