Calculus I Practice Problems with Limits and Derivatives

[dekoi]

1.) lim x --> 1(left) (x^2 + |x| -2)/(|1-x|)

Since the limit is from the left, i made all absolute values pnegative, therefore numerator = x^2 - x - 2, and similarly denom. = (x-1). Then, by inspection, the limit would equal to + infinite.

2.) Find numbers a and c such that lim x --> 0 x/(sqrt(ax + c) -3) = 2

Multiplying first by the conjugate, and then assuming a value for either a or c (in my case, i let c = 9). Then, solving for a, i got 3. Is this assumption allowed/necessary?

3.) lim (x-->0) f(x^2)/x^2, where f(0)=0 and f'(0)=3.

No idea.

4.) Evaluate f'(P), where f(P) = tan(3P + sinP))

No idea.

5.) Prove: If lim x--> 6 f(x)g(x) exists, then limit must be f(6)g(6).

6.) Prove: If f is continuous at 5 and f(5)=2 and f(4)=3, then lim x-->2f(4x^2 - 11) =2


Thank you for absolutely any input. These are only a selected few (out of very many) that i was unsuccessful at solving.

Thank you again.

 

[Leong]

#1 :
1. the numerator is x^2 + x -2 because when x approaches 1 from the left, it is a positive number; and can factorized.
2. simplify and i get -3.

 

[Warr]

4) Have you learned derivatives yet?

some things that may help you solve the problem

a) tan(x) = sin(x)/cos(x)
b) The two most important 'differentiation rules' here are the quotient and chain rule.
c) d/dx(sinx) = cosx, d/dx(cosx) = -sinx where d/dx is the derivative with respect to x if you haven't seen that notation before

Try posting anything you come up with