Precalculus continuity problem, use the 3 step definition of continuity to discuss the continuity (question is for jimthompson5910)
Answers
As the name implies, there are 3 parts or conditions that must be held true in order for the function to be continuous at x = a.
Those three conditions are:
f(a) existsThe limit [tex]\displaystyle \lim_{x\to a} f(x)[/tex] existsThe limiting value and the function value are the same, ie, [tex]\displaystyle \lim_{x\to a} f(x) = f(a)[/tex]---------------------------
In this case, a = -1 as this is where the junction happens. It's where f(x) swaps identity from the first piece x^2-3 to the second piece 3x+1
To compute f(a), aka f(-1), we'll use the second piece
f(x) = 3x+1
f(-1) = 3(-1)+1
f(-1) = -2
We see that f(a) does indeed exist, so condition (1) is held true.
----------------------------
To compute the limit, we'll need the left hand limit (LHL) and right hand limit (RHL)
The LHL is found by having x approach -1 from the left. So you'll start with something like x = -2, then move to x = -1.5 then to x = -1.1 then to x = -1.01, and so on, getting steadily closer to x = -1. We won't actually arrive at x = -1 itself.
But because x^2-3 is a polynomial, and all polynomials are continuous, we can simply plug x = -1 into this to find that...
y = x^2-3
y = (-1)^2 - 3
y = -2
The input x = -1 leads to the output y = -2 for the first piece. This is the LHL.
We found earlier that x = -1 lead to y = -2 for the second piece. This is the RHL.
Because LHL = RHL, we have proven condition (2) is true. It also means condition (3) is true because f(a) is part of the RHL, more or less.
----------------------------
In layman's terms, we can think of the two curves as roads. For the function to be continuous, the road cannot have any jumps, gaps, or potholes. Condition (1) says that the point must exist, aka there isn't a pothole there. Condition (2) says that the two pieces of the road, on either side of the point in question, must connect together. Hence there are no gaps or jumps. Condition (3) effectively ties everything together.
You might be asking if condition (3) is a bit redundant. Surely if f(a) exists and the limit exists, then that's enough to prove continuity, right? Unfortunately no that's not the case. Consider the situation where the limit exists, but f(a) was some other value. That means we have a removable discontinuity. That point in the road is a pothole but the two roads do connect.
One could argue that conditions (2) and (3) are sufficient, and condition (1) isn't really needed. This is because if condition (3) was the case, then we automatically have shown that f(a) must exist. This is of course assuming we found that the limit exists as well, and it's not plus/minus infinity.
Related Questions
Mrs. Franklin sells MSU hats and sweatshirts. She is hoping to earn $500 by charging
$20 for each sweatshirt and $10 for each hat. If she sells a total of 30 items, Mr.
Sanders wants to know how many of each item she sells.
Time
Atten
30N
Let H represent the number of hats and S represent the number of sweatshirts.
What equation relates S and H to the total Money earned?
205 + 10H = 500
105 + 20H - 500
105 = 500
10H - 400
Answers
Answer:
The answer is 20s+10h=500
Step-by-step explanation:
The reason is because 20 times 20=400 and then 10 times 10= 100 then you add those together and get $500
if the product of any two rational numbers is 2 and one of them is 1/7, find the other
Answers
Answer:
14
Step-by-step explanation:
1/7*X=2
X=2/(1/7)
X=2*7
X=14
➪ Product of two rational numbers = 2
➪ One of the number = [tex] \frac{1}{7} [/tex]
To find:-➪ The other number.
[tex]\huge{\boxed{\boxed{\tt{\purple{Solution\:⤵}}}}}[/tex]
[tex]\sf\blue{The \:other \:number \:is\: 14.}[/tex] ✅
Step-by-step explanation:-❥ Let the other number be [tex]x[/tex].
❥ As per the question, we have
[tex]product \: of \: two \: rational \: numbers \: = 2 \\ ⇢ \frac{1}{7} \times x = 2 \\ ⇢ x = 2 \times 7 \\ ⇢ x = 14[/tex]
[tex]\sf\red{Therefore,\:the\:other\: number\: is\: 14.}[/tex]
To verify:-[tex] \frac{1}{7} \times 14 = 2 \\➪ \: \frac{14}{7} = 2 \\ ➪ \: 2 = 2\\ ➪ \: L. H. S. = R. H. S.[/tex]
Hence verified.
[tex]\circ \: \: { \underline{ \boxed{ \sf{ \color{green}{Happy\:learning.}}}}}∘[/tex]
Please Help! Answer full step by step.
Triangle ABC has vertices A(- 3, - 4), B(16, - 2) and C(13, - 10) . Show algebraically that ABC is a right angled triangle.
Answers
Triangle ABC has the three sides AB, BC, and AC.
Let's find the slope of each line. We'll use the slope formula
Let's start with the slope of line AB.
A = (-3, -4) = (x1,y1)
B = (16, -2) = (x2,y2)
m = slope
m = (y2-y1)/(x2-x1)
m = (-2-(-4))/(16-(-3))
m = (-2+4)/(16+3)
m = 2/19
The slope of line AB is 2/19. I'll keep it in fraction form.
We'll use this later, so let p = 2/19.
---------------
Repeat those steps to find the slope of BC
B = (16, -2) = (x1,y1)
C = (13, -10) = (x2,y2)
m = (y2-y1)/(x2-x1)
m = (-10-(-2))/(13-16)
m = (-10+2)/(13-16)
m = -8/(-3)
m = 8/3
The slope of line BC is 8/3.
Let q = 8/3 so we can use it later.
---------------
Now find the slope of line AC.
A = (-3, -4) = (x1,y1)
C = (13, -10) = (x2,y2)
m = (y2-y1)/(x2-x1)
m = (-10-(-4))/(13-(-3))
m = (-10+4)/(13+3)
m = -6/16
m = -3/8
The slope of line AC is -3/8
Let r = -3/8.
---------------
To recap everything so far, we have
p = slope of line AB = 2/19q = slope of line BC = 8/3r = slope of line AC = -3/8We can see that,
p*q = (2/19)*(8/3) = 16/57q*r = (8/3)*(-3/8) = -1p*r = (2/19)*(-3/8) = -6/152 = -3/76Of those slope multiplications, q and r multiply to -1 is what we should focus on. If two slopes multiply to -1, then their lines are perpendicular. This applies to any perpendicular lines where neither line is vertical, and neither is horizontal.
The fractions 8/3 and -3/8 are negative reciprocals of one another. We flip the fraction and flip the sign from positive to negative, or vice versa. This is another way to see we have perpendicular slopes.
So in short, we've shown that BC and AC are perpendicular, and therefore triangle ABC is a right triangle.
----------------------------------------------------------------
To solve this problem a completely different way, we could follow this outline:
Step 1) Find the length of each side AB, BC, and AC. Use the distance formula. Step 2) Apply the pythagorean theorem converse. If you can show a^2+b^2 = c^2 is true, then the triangle with sides a,b,c is a right triangle, where c is the longest side (hypotenuse) opposite the right angle.Through step 1, you should find that,
AB = sqrt(365)BC = sqrt(73)AC = sqrt(292)Let a = sqrt(73), b = sqrt(292), c = sqrt(365). You should find that a^2+b^2 = c^2 is a true statement based on those values, and therefore we have a right triangle. The longest side is AB = sqrt(365), making angle C to be the 90 degree angle. The diagram is shown below. I used GeoGebra to make the diagram.
Renee wrote the pattern shown below. What is the rule for following Renee's pattern? 23, 47, 95, 191 A. multiply by 2 B.multiply by 2, add 1 C.multiply by 2, add 2 D.multiply by 2, subtract 1
Answers
Answer:
B. Multiply by 2, add 1
explanation:
23× 2 = 46 + 1 = 47
47 × 2 = 94 + 1 = 95
95 × 2 = 190 + 1 = 191
Maria practiced for her piano recital each day for three days. The first day she practiced for hour, the second day she practiced for hours, and the third day she practiced for hours. By how much did she increase the time she practiced each day? Group of answer choices
Answers
Answer:
3/4
Step-by-step explanation:
Maria practiced for her piano recital each day for three days. the first day she practiced for 3/4 hour, the second day she practiced for 1 1/2 hours, and the third day she practiced for 2 1/4 hours. by how much did she increase the time she practiced each day?
the increase in hours practiced can be determined by calculating the difference between the hours practiced on the second and first day
[tex]1\frac{1}{2} - \frac{3}{4}[/tex]
[tex]1\frac{2 - 3}{4}[/tex] the lowest common multiple that can divide 2 and 4 is 4.
[tex]\frac{6-3}{4}[/tex] if 2 is subtracted from 3, the answer would be negative. So, borrow 1 from the whole number. the value of this borrowed one would be equal to 4 which is the LCM. Add 4 to 2. this gives 6
[tex]\frac{3}{4}[/tex]
A movie theater has 85 seats. One rainy day in the summer, all seats were sold and the ticket income was $121. If adult tickets cost $3 and children tickets cost $1, how many children were in the audience?
Answers
Answer:
children = 67
Step-by-step explanation:
two equations can be derived from this question
x + y = 85 equation 1
3x + y = 121 equation 2
where x = number of adults
y = number of children
subtract equation 1 from 2
2x = 36
x = 18
substitute for x in equation 1
y = 85 - 18 = 67
Liz earns a salary of $2,100 per month, plus a commission of 3% of her sales. She wants to earn at least $2,900 this month. Enter an inequality to find amounts of sales that will meet her goal. Identify what your variable represents. Enter the commission rate as a decimal
Answers
Answer:
x≥26667
Step-by-step explanation:
let x= her sales
so the money she gets from her sales is .03x
if we wants at least 2900 we can write
2100+.03x≥2900
Solve for x
.03x≥800
x≥26666.6667
round up (because you can make a fraction of a sale) to get
x≥26667
what is the volume of a cone with a radius of 13cm and diameter 24cm?
Answers
4247.43327 Or short 4247.4!
Step-by-step explanation:
A movie theater designs two bags to hold 250 cubic inches of popcorn each.
Find the height of each bag.
Pleaseee helpp
Answers
Answer:
5 inches
Step-by-step explanation:
Well two bags hold 250 cubic inches, which means one bag will hold 125 cubic inches.
Then you cube route 125 because its cubic inches.
[tex]\sqrt[3]{125}[/tex] = 5
So I would think its five inches but I may be wrong <3
Can someone pls give me the answer to this?
Answers
Answer:
This is all the above
Step-by-step explanation:
all answers are true to this graph
What is the value of X in the triangle shown?
Answers
Answer:
Step-by-step explanation:
take 45 degree as reference angle
using sin rule
sin 45=opposite/hypotenuse
1[tex]\sqrt{2}[/tex]=x/16
x=16/[tex]\sqrt{2}[/tex]
x=8[tex]\sqrt{2}[/tex]
Jina has scored 79, 85, 89 and 68 on her previous four tests. What score does she need on her next test so that her mean is 80?
Answers
Answer:
79
Step-by-step explanation:
79, 85, 89, 68 have a mean of 80.25 according to calculator.net. the number itself Is close to the number you want, in this case 80, i input 80, but it gave me a score of 80.2, so i tried again with 81, it gave me a score of 80.4, so the third attempt was successful with 79 being the input and 80 the output
calculator.net is approved on my school computer so there is a chance it is on yours as well.
Which statement is true? Only some rectangles are parallelograms. Parallelograms have 2 pairs of parallel sides. Therefore, only some rectangles have 2 pairs of parallel sides. All rectangles are parallelograms. Parallelograms have exactly 1 pair of parallel sides. Therefore, all rectangles have exactly 1 pair of parallel sides. Only some rectangles are parallelograms. Parallelograms have exactly 1 pair of parallel sides. Therefore, only some rectangles have exactly 1 pair of parallel sides. All rectangles are parallelograms. Parallelograms have 2 pairs of parallel sides. Therefore, all rectangles have 2 pairs of parallel sides.
Answers
Answer:
STATEMENT NO. 4 IS CORRECT.
Step-by-step explanation:
Because every rectangle has two pair of parallel lines so it can be called a parallelogram.
x + y=100 x-y=10 pair of equation
Answers
Answer:
x = 55, y = 45
Step-by-step explanation:
add both equations: 2x = 110, x= 55, input back in to get y = 45
This is Right answer.....
I hope you understand....
Thanks....
Camille goes to sleep at 8:30 pm and wakes up at 6:00 am. How long does Camille sleep?
Answers
Answer:
9. 5 Hours
Step-by-step explanation:
8:30 AM to 12:00 PM = 3.5 hours
12:00 PM to 6:00 PM = 6 hours
Total = 6 + 3.5 = 9.5 hours
Hope this answer helps you :)
Have a great day
Mark Brainliest
Kendall makes candles and sells them online. To make an Essential Elegance candle, she adds
3 drops of lavender and 6 drops of vanilla to a bowl of unscented wax. To make a Garden
Glory candle, she adds 4 drops of lavender and 10 drops of vanilla to the bowl instead. Which
candle has a greater ratio of lavender to vanilla?
Answers
Answer:
Essential Elegance Candle
Step-by-step explanation:
Ratios are Lavender:Vanilla
For the Essential elegance candle the ratio is 3:6 which can be simplified to 1:2 whereas the Glory Candle is 4:10 which can be simplified to 2:5 on either one the vanilla is the bigger number but in terms of ratio it would be the essential elegance since 1/2>2/5
e) If A = {x:xis a letter in the word 'MATHEMATICS'}, list the members and find
n (A).
Answers
Given :
If, A = { x : x is a letter in the word 'MATHEMATICS' } .
To Find :
The members of A and find n (A).
Solution :
We have given a set, A = { x : x is a letter in the word 'MATHEMATICS' } .
Now, we know in set theory recurring element is written once only i.e all unique elements should be there in the set.
So,
A = { M, A, T, H, E, I, C, S }
Now, number of element in set A is 8.
Therefore, n(A) = 8 .
1) Is it a function?
Please help
Answers
Answer:
Yes it is a function
Hope its help you ☺
y>-4x+5 and y>2x-7, how to graph it?
Answers
Answer:
Graph.
y>−4x+5 and y>2x−7
Step-by-step explanation:
If ABC is reflected across the y-axis, what are the coordinates of A?
Answers
Answer:
A. (2,5)
Step-by-step explanation:
Answer:
C
Step-by-step explanation:
Solve the equation
-4(2-1.2x) = 40
Solve the equation
9=6y+33
Answers
Answer:
x = 10
y = -4
Step-by-step explanation:
[tex]\ -4(2 - 1.2x ) = 40\\\ -8 + 4.8x = 40\\\ 4.8x = 40+8\\\ 4.8x = 48\\x = \frac{48}{4.8} = \frac{48 \times 10}{48} = 10[/tex]
[tex]9 = 6y + 33\\6y = 9 -33\\6y = -24\\y = -4[/tex]
Arrange the matrices in increasing order of their determinant values.
Answers
Answer:
c, a, b
Step-by-step explanation:
Given
See attachment for matrices
(a)
[tex]D = \left[\begin{array}{cc}\cos\theta &\sin\theta\\ -\sin\theta& \cos\theta\\\end{array}\right][/tex]
The determinant of the matrix is:
[tex]|D |= (\cos\theta * \cos\theta - \sin\theta *- \sin\theta)[/tex]
[tex]|D | = \cos^2\theta + \sin^2\theta[/tex]
Using trigonometry ratio, we have:
[tex]|D | = 1[/tex]
(b)
[tex]\left[\begin{array}{ccc}2&0\\0&2\end{array}\right][/tex]
The determinant of the matrix is:
[tex]|D| = 2 * 2 - 0 * 0[/tex]
[tex]|D| = 4 - 0[/tex]
[tex]|D| = 4[/tex]
(c)
[tex]\left[\begin{array}{ccc}0&i\\-i&0\end{array}\right][/tex]
The determinant of the matrix is:
[tex]|D| = 0 * 0 -(-i * i)[/tex]
[tex]|D| = 0 +i^2[/tex]
[tex]|D| = i^2[/tex]
In complex numbers
[tex]i^2 = -1[/tex]
So:
[tex]|D| = -1[/tex]
So, the order of the determinants is: c, a, b
help please need done today
Answers
Answer:
77 ft.
Step-by-step explanation:
7*8+7*3=77
HELP ME THANKS +×+
Applying the Pythagorean theorem, Find the measure of the hypotenuse of a right triangle with legs of 11 inches and 60 inches.
Answers
Answer:
[tex]hypotenuse = \sqrt{ {height}^{2} + {adjacent}^{2} } \\ = \sqrt{ {11}^{2} + {60}^{2} } \\ = \sqrt{3721} \\ = 61 \: inches[/tex]
hypotenuse=
height
2
+adjacent
2
=
11
2
+60
2
=
3721
=61inches
I need help can you guys help me
Answers
Answer:
90
Step-by-step explanation:
a right angle is always 90 degrees
Explanation- because a triangle has to =180 degrees so if you have one right triangle (90 degrees) the other two angles have to add together to make 90.
Which statement make correctly uses limits determine a vertical asymptote of G(x)=-7(x-5)^2(x+6)/(x-5)(x+5)
Answers
Answer:
There is a vertical Asymptote at x = 5 because [tex]\lim_{x \to 5^{-} } G(x) = \infty\\lim_{x \to 5^{+} } G(x) = -\infty[/tex]
There is a vertical Asymptote at x = -5 because [tex]\lim_{x \to -5^{-} } G(x) = \infty\\\lim_{x \to -5^{+} } G(x) = -\infty[/tex]
Step-by-step explanation:
The exact question is as follows :
Given - G(x) = -7(x-5)^2(x+6)/(x-5)(x+5)
To find - Which statement make correctly uses limits determine a vertical Asymptote of G(x)
Solution -
Vertical Asymptotes are vertical lines which correspond to the zeroes of the denominator of a rational function.
Here the given function is the rational function
Denominator of G(x) = (x - 5)(x + 5)
So,
Put denominator = 0, we get
(x - 5)(x + 5) = 0
⇒x = 5, -5
∴ we get
G(x) has vertical Asymptotes at x = 5 and x = -5
Now,
At x= 5
[tex]\lim_{x \to 5^{-} } G(x) = \infty\\\lim_{x \to 5^{+} } G(x) = -\infty[/tex]
And
At x = -5
[tex]\lim_{x \to -5^{-} } G(x) = \infty\\\lim_{x \to -5^{+} } G(x) = -\infty[/tex]
∴ we get
There is a vertical Asymptote at x = 5 because [tex]\lim_{x \to 5^{-} } G(x) = \infty\\lim_{x \to 5^{+} } G(x) = -\infty[/tex]
There is a vertical Asymptote at x = -5 because [tex]\lim_{x \to -5^{-} } G(x) = \infty\\\lim_{x \to -5^{+} } G(x) = -\infty[/tex]
Answer:
Step-by-step explanation: A
For which function does f decrease by 15% every time x increase by 1? A: f(x)= 0.15^x B: f(x)=8.5^x C: f(x)=15^x. D: f(x)=85^x
Answers
Answer:
[tex]f(x) = (0.85)^x[/tex]
Step-by-step explanation:
Exponential amount of decay:
The equation for an amount that decays exponentially has the following format:
[tex]A(x) = A(0)(1 - r)^x[/tex]
In which r is the decay rate, as a decimal.
Decrease by 15%
This means that [tex]r = 0.15[/tex]. So
[tex]f(x) = (1 - r)^x = (1 - 0.15)^x = (0.85)^x[/tex]
So
[tex]f(x) = (0.85)^x[/tex]
A Skateboard is reduced 25%
in price in a sale. The old price
was $122. Find the new price.
Answers
Answer: 30.5
Step-by-step explanation:
Which is the solution of the system of equations shown in the graph?
Answers
Answer:
(5, -7).
Step-by-step explanation:
Solution is the point of intersection.
(5, -7).
Beatrice made 7 2/3 cups of lemonade for her lemonade stand. Later, she made 10 3/5 more cups of lemonade. How many cups of lemonade did she make in total?
A: 17 4/15
B: 17 5/8
C: 18 4/15
D: 5/8
Answers
Answer:
C: 18 4/15
Step-by-step explanation:
7 2/3 + 10 3/5
Convert fractions so that they share a common denoninator:
7 10/15 + 10 9/15
17 19/15
18 4/15