1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
iren2701 [21]
1 year ago
11

Calculus piecewise function. ​

Mathematics
1 answer:
Kipish [7]1 year ago
4 0

Part A

The notation \lim_{x \to 2^{+}}f(x) means that we're approaching x = 2 from the right hand side (aka positive side). This is known as a right hand limit.

So we could start at say x = 2.5 and get closer to 2 by getting to x = 2.4 then to x = 2.3 then 2.2, 2.1, 2.01, 2.001, etc

We don't actually arrive at x = 2 itself. We simply move closer and closer.

Since we're on the positive or right hand side of 2, this means we go with the rule involving x > 2

Therefore f(x) = (x/2) + 1

Plug in x = 2 to find that...

f(x) = (x/2) + 1

f(2) = (2/2) + 1

f(2) = 2

This shows \lim_{x \to 2^{+}}f(x) = 2

Then for the left hand limit \lim_{x \to 2^{-}}f(x), we'll involve x < 2 and we go for the first piece. So,

f(x) = 3-x

f(2) = 3-2

f(2) = 1

Therefore, \lim_{x \to 2^{-}}f(x) = 1

===============================================================

Part B

Because \lim_{x \to 2^{+}}f(x) \ne \lim_{x \to 2^{-}}f(x) this means that the limit \lim_{x \to 2}f(x) does not exist.

If you are a visual learner, check out the graph below of the piecewise function. Notice the gap or disconnect at x = 2. This can be thought of as two roads that are disconnected. There's no way for a car to go from one road to the other. Because of this disconnect, the limit doesn't exist at x = 2.

===============================================================

Part C

You'll follow the same type of steps shown in part A.

However, keep in mind that x = 4 is above x = 2, so we'll deal with x > 2 only.

So you'd only involve the second piece f(x) = (x/2) + 1

You should find that f(4) = 3, and that both left and right hand limits equal this value. The left and right hand limits approach the same y value. The limit does exist here. There are no gaps to worry about when x = 4.

===============================================================

Part D

As mentioned earlier, since \lim_{x \to 4^{+}}f(x) = \lim_{x \to 4^{-}}f(x) = 3, this means the limit \lim_{x \to 4}f(x) does exist and it's equal to 3.

As x gets closer and closer to 4, the y values are approaching 3. This applies to both directions.

You might be interested in
Kim needs to place 32 mystery books and 56 biography books on library shelves. She is to place the maximum number of books on ea
EleoNora [17]
8 books will be on each shelf. There will be 4 shelves for the mystery books and 7 shelves for the biography books.
3 0
3 years ago
Which is the solution set of the inequality<br> -15y+9&lt;-36
liraira [26]

Answer:

-15y+93\:\\ \:\mathrm{Interval\:Notation:}&\:\left(3,\:\infty \:\right)\end{bmatrix}

Therefore, the first choice is correct.

The graph of the inequality is also attached below.

Step-by-step explanation:

Considering the inequality

-15y+9

solving

-15y+9

\mathrm{Subtract\:}9\mathrm{\:from\:both\:sides}

-15y+9-9

-15y

\mathrm{Multiply\:both\:sides\:by\:-1\:\left(reverse\:the\:inequality\right)}

\left(-15y\right)\left(-1\right)>\left(-45\right)\left(-1\right)

15y>45

\mathrm{Divide\:both\:sides\:by\:}15

\frac{15y}{15}>\frac{45}{15}

y>3

In other words,

-15y+93\:\\ \:\mathrm{Interval\:Notation:}&\:\left(3,\:\infty \:\right)\end{bmatrix}

Therefore, the first choice is correct.

The graph of the inequality is also attached below.

6 0
3 years ago
What two numbers multiply to get 45 but add to get -10?
goldenfox [79]
There are no two numbers that multiply to 45 but add to -10.
This is because the only numbers that multiply to get 45 are 1x45, 3x15, 5x8. None of them can add to -10 even if you make some of the numbers negative instead.

If this is a factoring problem, you may have to use the quadratic equation instead.
7 0
2 years ago
URGENT: PLEASE HELP!
vazorg [7]

To solve this problem, we are going to set up an equation. However, first we need to convert the percentages to decimals by dividing by 100.

15% = 0.15 , 7.5% = 0.075

Now, we should substitute in our given information into our equation and let our unknown value be equal to the variable x.

0.15(500) = 0.075x

Then, we should simplify our equation by multiplying together the two factors on the left side of the equation.

75 = 0.075x

Finally, we should divide both sides of the equation by 0.075 to get the variable x alone.

x = 1000

Therefore, your answer is 1000.

Hope this helps!

4 0
3 years ago
Read 2 more answers
What is the slope of the line whose equation is y - = 0? slope = slope = - slope = 0
sasho [114]

Answer:

the slope is 0

Step-by-step explanation:

7 0
3 years ago
Other questions:
  • J.R. tosses a coin 100 times, resulting in it landing on heads 40 times and tails 60 times. Based on his result, what is the exp
    6·2 answers
  • How do you simplify 520/480
    7·1 answer
  • Please Help me. I Need help fast. Only answer if you can explain your work.
    8·2 answers
  • Consider the function f(x)=−2x3+2x2−2x+2 Find the average slope of this function on the interval (4,9). By the Mean Value Theore
    10·1 answer
  • You have $20 to spend on taxi fare. The ride costs $5 plus $2.50 per kilometer.
    15·1 answer
  • There are 10 kids that want ice cream. Each one can choose between chocolate, vanilla or strawberry. You'll collect their choice
    10·1 answer
  • Pleass answer the following​
    5·1 answer
  • What is 55/99 Rounded to the nearest half
    15·2 answers
  • 11. Madison wants to know which sports are most popular.
    13·2 answers
  • U ARE SO SMART IF YOU ANSWER AND YOU ARE SO BRAINLY. PLZ HLEP ASap
    14·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!