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
kicyunya [14]
3 years ago
7

Bsinx%5E%7B2%7D%20%7D" id="TexFormula1" title="\lim_{x\to \ 0} \frac{\sqrt{cos2x}-\sqrt[3]{cos3x} }{sinx^{2} }" alt="\lim_{x\to \ 0} \frac{\sqrt{cos2x}-\sqrt[3]{cos3x} }{sinx^{2} }" align="absmiddle" class="latex-formula">
Mathematics
1 answer:
salantis [7]3 years ago
6 0

Answer:

\displaystyle  \lim_{x \to 0} \frac{\sqrt{cos(2x)} - \sqrt[3]{cos(3x)}}{sin(x^2)} = \frac{1}{2}

General Formulas and Concepts:

<u>Calculus</u>

Limits

Limit Rule [Variable Direct Substitution]:                                                                     \displaystyle \lim_{x \to c} x = c

L'Hopital's Rule

Differentiation

  • Derivatives
  • Derivative Notation

Basic Power Rule:

  1. f(x) = cxⁿ
  2. f’(x) = c·nxⁿ⁻¹

Derivative Rule [Chain Rule]:                                                                                    \displaystyle \frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)

Step-by-step explanation:

We are given the limit:

\displaystyle  \lim_{x \to 0} \frac{\sqrt{cos(2x)} - \sqrt[3]{cos(3x)}}{sin(x^2)}

When we directly plug in <em>x</em> = 0, we see that we would have an indeterminate form:

\displaystyle  \lim_{x \to 0} \frac{\sqrt{cos(2x)} - \sqrt[3]{cos(3x)}}{sin(x^2)} = \frac{0}{0}

This tells us we need to use L'Hoptial's Rule. Let's differentiate the limit:

\displaystyle  \lim_{x \to 0} \frac{\sqrt{cos(2x)} - \sqrt[3]{cos(3x)}}{sin(x^2)} = \displaystyle  \lim_{x \to 0} \frac{\frac{-sin(2x)}{\sqrt{cos(2x)}} + \frac{sin(3x)}{[cos(3x)]^{\frac{2}{3}}}}{2xcos(x^2)}

Plugging in <em>x</em> = 0 again, we would get:

\displaystyle \lim_{x \to 0} \frac{\frac{-sin(2x)}{\sqrt{cos(2x)}} + \frac{sin(3x)}{[cos(3x)]^{\frac{2}{3}}}}{2xcos(x^2)} = \frac{0}{0}

Since we reached another indeterminate form, let's apply L'Hoptial's Rule again:

\displaystyle \lim_{x \to 0} \frac{\frac{-sin(2x)}{\sqrt{cos(2x)}} + \frac{sin(3x)}{[cos(3x)]^{\frac{2}{3}}}}{2xcos(x^2)} = \lim_{x \to 0} \frac{\frac{-[cos^2(2x) + 1]}{[cos(2x)]^{\frac{2}{3}}} + \frac{cos^2(3x) + 2}{[cos(3x)]^{\frac{5}{3}}}}{2cos(x^2) - 4x^2sin(x^2)}

Substitute in <em>x</em> = 0 once more:

\displaystyle \lim_{x \to 0} \frac{\frac{-[cos^2(2x) + 1]}{[cos(2x)]^{\frac{2}{3}}} + \frac{cos^2(3x) + 2}{[cos(3x)]^{\frac{5}{3}}}}{2cos(x^2) - 4x^2sin(x^2)} = \frac{1}{2}

And we have our final answer.

Topic: AP Calculus AB/BC (Calculus I/I + II)

Unit: Limits

You might be interested in
Given the figure, find the values of x and z.
matrenka [14]

Answer:

z = 62 \\ 6x + 52 = 180 - 62 \\ 6x = 66 \\ x = 11

7 0
2 years ago
Read 2 more answers
Can someone explain to me how to do this? I just don't get it.
Maksim231197 [3]

Answer:

x=130

Step-by-step explanation:

triangle=180

180-34-96=50

line=180

180-50=130

7 0
2 years ago
4x:3 = 6:5 calculate the value of x
My name is Ann [436]

Answer:

-2

Step-by-step explanation:

4x:3=6:5.

4x/3=6/5

4x × 5=6×3

20 X=18

X=18-20

X=-2

4 0
2 years ago
The student body president of a high school claims to know the names of at least 1000 of the 1800 students who attend the school
natali 33 [55]

Answer: Population = Total students attend the school.

Parameter = Proportion of students named by president correctly.

Step-by-step explanation:

Population for a particular study is the largest possible group of individuals that are essential to be considered.

Parameter = measure of characteristic in a population.

By the above definition, we have

Population = Total students attend the school(1800).

Parameter = Proportion of students named by president correctly.

8 0
2 years ago
What is the measure of the vertex angle
maw [93]

Answer:

Vertex angle = 124°

Step-by-step explanation:

m<BAC = m<BCA (base angles of isosceles ∆ are congruent)

7x + 1 = 5x + 9

Collect like terms

7x - 5x = -1 + 9

2x = 8

divide both sides by 2

x = 4

Vertex angle = 180 - ((7x + 1) + (5x + 9)) (sum of triangle)

Plug in the value of x

Vertex angle = 180 - ((7*4 + 1) + (5*4 + 9))

Vertex angle = 180 - (29 + 29)

Vertex angle = 124°

5 0
3 years ago
Other questions:
  • What is the area of the polygon below
    7·2 answers
  • A rectangle has an area of 102 cm ^2. The length of the rectangle is 17 cm.
    8·1 answer
  • PLEASE HURRY WILL GIVE BRAINLIEST!!<br><br> What is the surface area of this rectangular prism?
    9·2 answers
  • How many snakeman and red crows books were sold altogether
    9·2 answers
  • HW
    5·1 answer
  • What is the coefficient of x2 in the expansion of (x + 2)^3
    13·1 answer
  • A cheetah can run 70 miles per hour what is this speed in feet
    9·1 answer
  • Write an equation for the following line.<br> 10<br> -10<br> 10<br> -101
    7·1 answer
  • Pls help this question i want its aregent
    11·1 answer
  • Solve the eequation show all your work g + 5 = 7 1/5
    6·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!