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
kompoz [17]
3 years ago
12

Hiiii.. please help me with this limit question ​

Mathematics
1 answer:
Alenkasestr [34]3 years ago
8 0

Answer:

π

Step-by-step explanation:

Solving without L'Hopital's rule:

lim(x→0) sin(π cos²x) / x²

Use Pythagorean identity:

lim(x→0) sin(π (1 − sin²x)) / x²

lim(x→0) sin(π − π sin²x) / x²

Use angle difference formula:

lim(x→0) [ sin(π) cos(-π sin²x) − cos(π) sin(-π sin²x) ] / x²

lim(x→0) -sin(-π sin²x) / x²

Use angle reflection formula:

lim(x→0) sin(π sin²x) / x²

Now we multiply by π sin²x / π sin²x.

lim(x→0) [ sin(π sin²x) / x² ] (π sin²x / π sin²x)

lim(x→0) [ sin(π sin²x) / π sin²x] (π sin²x / x²)

lim(x→0) [ sin(π sin²x) / π sin²x] lim(x→0) (π sin²x / x²)

π lim(x→0) [ sin(π sin²x) / π sin²x] [lim(x→0) (sin x / x)]²

Use identity lim(u→0) (sin u / u) = 1.

π (1) (1)²

π

Solving with L'Hopital's rule:

If we plug in x = 0, the limit evaluates to 0/0.  So using L'Hopital's rule:

lim(x→0) [ cos(π cos²x) (-2π cos x sin x) ] / 2x

lim(x→0) [ -π cos(π cos²x) sin(2x) ] / 2x

-π/2 lim(x→0) [ cos(π cos²x) sin(2x) ] / x

Again, the limit evaluates to 0/0.  So using L'Hopital's rule one more time:

-π/2 lim(x→0) [ cos(π cos²x) (2 cos(2x)) + (-sin(π cos²x) (-2π cos x sin x)) sin(2x) ] / 1

-π/2 lim(x→0) [ 2 cos(π cos²x) cos(2x) + π sin(π cos²x) sin²(2x) ]

-π/2 (-2)

π

You might be interested in
It costs Linus a processing fee of $45 to set up a new cell phone account with AT&T plus $65 per month for the cell phone pl
valkas [14]

Answer: 110-35=x

answer: 7 minutes

I came here to find the answer to this proble but since no one answered I’ll just put the answer that I got.

Step-by-step explanation:

7 0
3 years ago
If a plate is moving 3.3 cm/yr, what distance [in kilometers (km)] will the plate move in 1 million years?
sasho [114]

\huge{\textbf{\textsf{{\color{pink}{An}}{\red{sw}}{\orange{er}} {\color{yellow}{:}}}}}

<h2>33 km</h2>

3.3cm = 0.000033km (divide by 100,000.

So, 0.000033km/yr.

0.000033 × 1,000,000

=33km

  • Thanks
  • Hope it helps

3 0
3 years ago
A ball is kicked 4 feet above the ground with an initial vertical velocity of 55 feet per second. The function h(t)=-16t^2+55t+4
Shalnov [3]

Answer:

Time in seconds when the ball is 4 feet above the ground is:

t= 0.6 and 2.9.

6 0
3 years ago
exercise 5 explain how you know the values of the digits in the number 58?then exercise 6 `explain why you should record a 1 in
melamori03 [73]
58 is 50 plus 8. So its 50 and 8.That is for the first one.
3 0
3 years ago
Use rounding to estimate 0.93+0.18 plz help me
Anastasy [175]
0.93 = 0.9
0.18 = 0.2 

0.9 + 0.2 = 1.1

Answer 1.1
6 0
3 years ago
Read 2 more answers
Other questions:
  • Expression is equivalent to 5(6x+3y)
    12·2 answers
  • Which is a recursive formula for the sequence 99.4, 0, –99.4, –198.8, where f(1) = 99.4?
    7·2 answers
  • Five different stores sell a bottle of water for one of the following prices:$1.45,$1.55,$1.85,$1.95,$1.45 what are the mean,med
    6·2 answers
  • Simplify 5^4/5. ~Please Hurry !!
    14·2 answers
  • What does m(x)=-1/4[-2]-1
    8·1 answer
  • Choose the best representation 8/11
    12·2 answers
  • Sum of 12/13+(-1/13)=
    10·2 answers
  • I NEED HELP WITH THIS QUESTION PLEASE ? :(
    13·1 answer
  • Below is data collected from a random sample of 80 students regarding
    9·1 answer
  • Points gyehdjskhrbedsx
    11·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!