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Chapter 1: Problem 32

Algebraically determine the limits. $$ \lim _{h \rightarrow 0} \frac{(3+h)^{2}-3^{2}}{h} $$

Short Answer

Expert verified
The limit is 6.

Step by step solution

01

Expand the Numerator

First, we need to expand \((3 + h)^2\) which is part of the numerator. Use the binomial formula to expand: \((3 + h)^2 = 9 + 6h + h^2\). Thus, the expression becomes: \[\frac{9 + 6h + h^2 - 9}{h}\] This simplifies the numerator to \(6h + h^2\).
02

Simplify the Fraction

Now, simplify the expression \(\frac{6h + h^2}{h}\). Factor \(h\) out of the numerator:\[\frac{h(6 + h)}{h}\] By cancelling \(h\) from the numerator and denominator, we are left with:\[6 + h\]
03

Calculate the Limit

With the simplified expression \(6 + h\), find the limit as \(h\) approaches 0:\[\lim_{h \to 0} (6 + h) = 6 + 0 = 6\]

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Key Concepts

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

Binomial Expansion
Binomial expansion is a technique used to expand expressions that are raised to a power, specifically binomials, which are algebraic expressions containing two terms. In our problem, we are dealing with the expression \( (3+h)^2 \). To expand this expression, we apply the formula for a binomial squared: \( (a + b)^2 = a^2 + 2ab + b^2 \).
Let's break it down step by step:
  • First, square the term 3: \(3^2 = 9\)
  • Next, find twice the product of 3 and h: \(2 \times 3 \times h = 6h\)
  • Finally, square the term h: \(h^2\)
Combine these results, and you'll get \(9 + 6h + h^2\). This expansion allows for the expression to become manageable for further operations, such as simplification or evaluation in limits.
Simplifying Fractions
Simplifying fractions is all about reducing them to their simplest form by eliminating common factors in the numerator and the denominator. In this exercise, after expanding the numerator, we obtained the expression \(\frac{9 + 6h + h^2 - 9}{h}\). This simplifies to \(\frac{6h + h^2}{h}\).
To simplify this fraction, we first factor out the common factor "h" from the numerator:
  • Factor \(h\) from \(6h + h^2\): \(h(6 + h)\)
By cancelling the "h" from both the numerator and the denominator, you're left with the simple expression "6 + h". This step is crucial in reducing the complexity of expressions, making it easier to evaluate in subsequent mathematical operations.
Calculating Limits
Calculating the limit of a function involves finding the value that a function approaches as the variable approaches a certain point. In calculus, it's a fundamental concept used in defining derivatives and integrals.
For this problem, after simplifying the expression to \(6 + h\), we need to calculate the limit as \(h\) approaches 0:
  • Substitute \(h = 0\) into the expression: \(6 + 0\)
  • The limit, therefore, is \(6\)
This calculation shows how the expression behaves as \(h\) becomes infinitesimally small. Understanding limits is essential, as it forms the foundation for many concepts in calculus and helps in analyzing the behavior of functions at certain points.

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Most popular questions from this chapter

In desert areas of the western United States, lizards and other reptiles are harvested for sale as pets. Because reptiles hibernate during the winter months, no reptiles are gathered during the months of January, February, November, and December. The number of lizards harvested during the remaining months can be modeled as $$h(m)=15.3 \sin (0.805 m+2.95)+16.7 $$where \(m\) is the month of the year (i.e., \(m=3\) represents$$\text { March), } 3 \leq m \leq 10$$ (Source: Based on information from the Nevada Division of Wildlife) a. Calculate the amplitude and average value of the model. b. Calculate the highest and lowest monthly harvests. Write a sentence interpreting these numbers in context. c. Calculate the period of the model. Is this model useful for \(m\) outside the stated input range? Explain.

For each data set, write a model for the data as given and a model for the inverted data. Water Pressure Pressure Under Water $$ \begin{array}{|c|c|} \hline \begin{array}{c} \text { Depth } \\ \text { (feet) } \end{array} & \begin{array}{c} \text { Pressure } \\ \text { (atm) } \end{array} \\ \hline \text { Surface } & 1 \\ \hline 33 & 2 \\ \hline 66 & 3 \\ \hline 99 & 4 \\ \hline 132 & 5 \\ \hline \end{array} $$

Powdered drink mixes are consumed more during warmer months and less during colder months. A manufacturer of powdered drink mixes estimates the sales of its drink mixes will peak near 800 million reconstituted pints during the 28 th week of the year and will be at a minimum near 350 million reconstituted pints during the 2 nd week of the year. a. Calculate the period and horizontal shift of the drink mix sales cycle. Use these values to calculate the parameters \(b\) and \(c\) for a model of the form \(f(x)=a \sin (b x+c)+d\) b. Calculate the amplitude and average value of the drink mix sales cycle. c. Use the constants from parts \(a\) and \(b\) to construct a sine model for drink mix sales.

Stock Car Racing The amount of motor oil used during the month by the local stock car racing team is \(g(t)\) hundred gallons, where \(t\) is the number of months past March of a given year. a. Write a sentence of interpretation for \(g(3)=12.90\). b. Write the function notation for the statement "In October, the local stock car racing team used 1,520 gallons of motor oil."

For each of the functions, mark and label the amplitude, period, average value, and horizontal shift. \(f(x)=0.1 \sin (4 x-2)-0.5\)
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