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
Please help me with this, its confusing
ser-zykov [4K]
I can’t even read it ...
step by step explanation:
3 0
3 years ago
Find the product.<br> 5x2.2y2
ozzi

Answer:10x^2y^2

Step-by-step explanation:

5x2 X 2y2=10x^2y^2

3 0
3 years ago
PLEASE HELP - image attachment below, thanks! :)
CaHeK987 [17]
32 as well according to some rule in geometry
8 0
2 years ago
Read 2 more answers
Y varies jointly as x and z, and y = 32 m, x = 6 m, and z = 8 m. what is the value of y when x = 5 m and z = 12 m?
Fynjy0 [20]
As x = 5 & z=12, 

<span>2/3 = y /(xz) = y/(5*12) </span>

<span>= y/60 </span>

<span>so y / 60 = 2/3 </span>

<span>y = (2*60)/3 </span>

<span>= 40</span>
4 0
3 years ago
Help plz. Ignore the orange around the choice.
VMariaS [17]

Answer:

May I suggest -11

Step-by-step explanation:

4 0
3 years ago
Other questions:
  • A basket of goods costs $85 in year 1. If the same basket costs $90 in year 2, what is the CPI for year 2?
    13·2 answers
  • What is the answer for (-6)(-3) can you you please walk me through it then answer. I think it’s 2 -2 -18 or 18 please help me!
    6·2 answers
  • AC=x+14
    6·1 answer
  • Jackson School's spelling bee started at
    11·1 answer
  • Briana has just finished her sixth grade scrapbook.in her scrapbook,47% of the pages include her twin sister,Bethany.The scrapbo
    14·2 answers
  • The total cost for 4 people to ride the bus if the fare of $0.75 is increased by d dollars??
    5·1 answer
  • If 6 out of 30 people over the age 100 are male, how many are female
    6·1 answer
  • Danielle earns a 8.25% commission on everything she sells at the electronics store where she works. She also earns a base salary
    13·1 answer
  • What 666.983 divided by 978.087 is?
    7·2 answers
  • In how many ways can 10 horses be arranged in a merry-go-round?​
    5·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!