L HOPITALS and derivative

Question

ive got 3 questions

 

lim x approaches positive 1 (sq rt of x^2 - 1) / (arcsecx)

 

lim x approaches 0  2arctan3x^2 / 7x^2

 

 

initial value prob

 

dy/dx=1/(x^2+1) -1

 

 

I NEED HELP ASAP TNX

Answer 1

lim_x→1 √(x² - 1) / sec^-1 x

 

= lim_x→1 [x/√(x² - 1)] / [1/√(x² - 1)]

 

= lim_x→1 x

 

= 1

 

 

lim_x→0 2tan^-1(3x²) / (7x²)

 

= lim_x→0 [12x/(1+9x⁴)] / (14x)

 

= lim_x→0 6/[7(1+9x⁴)]

 

= 6/7

 

 

For the initial value problem, it is not clear from what you type, what the initial values of x and the value y there.

 

Assuming the -1 is part of the equation, the general solution is:

 

y = tan^-1 x - x + C.

 

If -1 was supposed to be part of the initial value, the general solution is just:

 

y = tan^-1 x + C.

 

So you should check your book for the initial value constraint and plug it into the general solution to get what the constant C should be.