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
valentinak56 [21]
3 years ago
8

Because of their connection with secant​ lines, tangents, and instantaneous​ rates, limits of the form ModifyingBelow lim With h

right arrow 0 StartFraction f (x plus h )minus f (x )Over h EndFraction occur frequently in calculus. Evaluate this limit for the given value of x and function f. f (x )equals StartRoot x EndRoot​, xequals11
Mathematics
1 answer:
Gre4nikov [31]3 years ago
8 0

Answer:

\dfrac{1}{2\sqrt{x}}

Step-by-step explanation:

f(x) = \sqrt{x} = x^{\frac{1}{2}}

f(x+h) = \sqrt{x+h} = (x+h)^{\frac{1}{2}}

We use binomial expansion for (x+h)^{\frac{1}{2}}

This can be rewritten as

[x(1+\dfrac{h}{x})]^{\frac{1}{2}}

x^{\frac{1}{2}}(1+\dfrac{h}{x})^{\frac{1}{2}}

From the expansion

(1+x)^n=1+nx+\dfrac{n(n-1)}{2!}+\ldots

Setting x=\dfrac{h}{x} and n=\frac{1}{2},

(1+\dfrac{h}{x})^{\frac{1}{2}}=1+(\dfrac{h}{x})(\dfrac{1}{2})+\dfrac{\frac{1}{2}(1-\frac{1}{2})}{2!}(\dfrac{h}{x})^2+\tldots

=1+\dfrac{h}{2x}-\dfrac{h^2}{8x^2}+\ldots

Multiplying by x^{\frac{1}{2}},

x^{\frac{1}{2}}(1+\dfrac{h}{x})^{\frac{1}{2}}=x^{\frac{1}{2}}+\dfrac{h}{2x^{\frac{1}{2}}}-\dfrac{h^2}{8x^{\frac{3}{2}}}+\ldots

x^{\frac{1}{2}}(1+\dfrac{h}{x})^{\frac{1}{2}}-x^{\frac{1}{2}}=\dfrac{h}{2x^{\frac{1}{2}}}-\dfrac{h^2}{8x^{\frac{3}{2}}}+\ldots

\dfrac{x^{\frac{1}{2}}(1+\dfrac{h}{x})^{\frac{1}{2}}-x^{\frac{1}{2}}}{h}=\dfrac{1}{2x^{\frac{1}{2}}}-\dfrac{h}{8x^{\frac{3}{2}}}+\ldots

The limit of this as h\to 0 is

\lim_{h\to0} \dfrac{f(x+h)-f(x)}{h}=\dfrac{1}{2x^{\frac{1}{2}}}=\dfrac{1}{2\sqrt{x}} (since all the other terms involve h and vanish to 0.)

You might be interested in
Just list all of the possible answers
Genrish500 [490]

Answer:

what grade are you??........

8 0
2 years ago
Read 2 more answers
Alyssa wants to buy a new car and is planning to take out a loan to pay for the car.
g100num [7]

Answer: the amount of interest that Alyssa will pay on her loan is $1320

Step-by-step explanation:

The formula for determining simple interest is expressed as

I = PRT/100

Where

I represents interest paid on the loan.

P represents the principal or amount taken as loan

R represents interest rate

T represents the duration of the loan in years.

From the information given,

P = $22000

R = 4.5%

T = 4 years

Therefore,

I = (22000 × 1.5 × 4)/ 100

I = 132000/100 = $1320

4 0
3 years ago
Cheryl plans to bake cookies tomorrow and needs to purchase chocolate chips and pecans for no more than $25 total. Chocolate chi
boyakko [2]

The inequality that correctly represents this scenario is given as follows:

2x + 5y \leq 25

The graph is given at the end of the question.

<h3>What is a system of inequalities?</h3>

A system of inequalities is when two or more variables are related, and inequalities are built to find the values of each variable.

In this problem, the variables are given as follows:

  • Variable x: number of chocolate chips purchased.
  • Variable y: number of pecans sold.

Chocolate chips are $2 per pound and pecans are $5 per pound. She wants to spend at most $25, hence:

2x + 5y \leq 25

More can be learned about a system of inequalities at brainly.com/question/3656398

#SPJ1

6 0
2 years ago
Find The Area Of The Shape Shown Below
Vesnalui [34]

Answer:

<u>32</u>

Step-by-step explanation:

I proceeded to divide the shape up into 2 triangle and 1 square. The added sum of the 2 divided triangles is 16 and the square is 16. Add those to to get <u>32</u>.

The following shape is a trapezoid

area formula:

A=

A=

A= (4)(12+4)

A= (4)(16)

A= (4)(16)

A= (64)

A=32

Also, is that Khan Academy?

8 0
3 years ago
Read 2 more answers
I need so much help!!!!
USPshnik [31]

<em>Answer:</em>

94.25m

<em>Explanation:</em>

\frac{3cm}{4.5m} × \frac{19.5}{x}

Solve for x.

(cross multiply & divide)

4.5 × 19.5 = 282.75

282.75 ÷ 3 = 94.25

4 0
3 years ago
Other questions:
  • What is the least common multiple for 3,5,and 7
    9·1 answer
  • Which one ????????????
    6·1 answer
  • Opposite of -(-24). Explain how you did it please so I get an ideo of what to do for the rest
    5·2 answers
  • Here's a graph of a linear function. Write the
    6·1 answer
  • Simplify the expression 2[13-1)^2/2]
    15·1 answer
  • Simplify, using the distributive property and then combining like terms.<br><br> 5(3a+b)−2(3a+b)
    13·2 answers
  • Um. i just need help with this! so if anyone knows help me plz &lt;3
    12·2 answers
  • PLEASE HELP AND ASAP!!!! look at screenshot (10 pts)
    12·2 answers
  • You roll a die 4 times. What is the probability of rolling a 3 on all 4 times?
    5·1 answer
  • Fill in the blank. Using the number line below, the plotted number is when rounded to the nearest thousand.​
    8·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!