Observing the fraction
(sin(t))^2 / (4t)^2
we see that both the numerator and the denominator approach 0 as t approaches 0. Thus the limit
lim(t->0) (sin(t))^2 / (4t)^2
is said to be "of indeterminate form 0/0", ...

Solve the integral -x dx with limits from 0 to x
I am actually solving the differential equation, y'=-xy y(0)=5, but I need to see the above integral solved from 0 to x to reassure myself that my other problem is working correctly.

Please could you solve the following question showing every stage as simply as possible to get to the correct answer.
Where any calculus rules are used could you please explain.
Integrate sec^2(3t) - cosec^2(5t) between the limits t = 0 and t = pi/4
Please see attached for a more clear version.

Use one-sided limits to find the limit or determine that the limit does not exist.
16-x^2 /4-x
lim x => 4
Find the trigonometric limit:
sin3x/2x
limx => 0
Please show work.

A medical facility does MRI's for sports injuries. Occasionally a test yields inconclusive results and must be repeated. Using the following sample data and n = 200, determine the upper and lower control limits for the fraction of retests using 2-sigma limits. Is the process in control?
Sample
1 2 3 4 5 6 7 8 9

Please show all work; don't explain each step. Please DON'T submit back as an attachment.Thank you
Sketch the solid bounded by the graphs of the given equation and find its volume by triple integration: z = y, y = x^2, y = 4, z = 0

See attachment
a. Use the definition of limits to show that
b. Compute the following limits (justify your answers).
1. ( )
2.
c. Let ( be a sequence. Show that ( is convergent if and only if the sequence ( and ( are convergent to the same limit.

For the following data use the weighting constant x = 0.5 and exponential smoothing to determine the forecast for 2005.
Subscribers Subscribers
Year (millions) Year (Millions)
1993 16.0 1999 86.0
1994 24.1 2000

11. Product filling weights for a cereal manufacturer are normally distributed with a mean of 350 grams, a variance of 225 grams and a standard deviation of 15 grams.
a) Develop the control limits that would be used on the company's x-bar control charts of samples of sizes 10, 20, 30.
b) What happens to the control limits a