evaluate the limit using l'hopital's rule (multiple choice)

Question

Hi, I have these multiple choice answers that I would like help on.

1. Evaluate the limit

lim (cot x - 1/x)

x→0

How should the given limit be​ evaluated? Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.

a. use l'hopital's rule once, rewrite the limit in its final form as lim ______________

b. use l'hopital's rule exactly once, to rewrite the limit as ________

c. multiply the expression by a unit fraction to obtain lim_______

d. use direct substitution

Evaluate the following limit

lim x²csc x

x→0

How should the given limit be​ evaluated? Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.

a. use direct substitution

b. multiply the expression by a unit fraction to obtain lim______

c. use l'hopital's rule exactly once to rewrite limit as lim__________

d. use l'hopital's rule more than once, rewrite the limit in its final form as lim__________

Thank you for your help!

Answer 1

cot x - 1/x =

cos x/sin x - 1/x =

(x cos x - sinx)/(x sinx) <-- indeterminant 0/0

L'hopitals:

(-xsinx + cosx - cos)/( xcosx + sinx) =

(-xsinx)/(x cos x + sinx) <--- indeterminant 0/0

L'hopitals again:

(-xcosx - sinx)/( -xsinx + cos x + cosx) =

(-xcosx - sinx) / (-xsinx + 2cos x)

denominator is now 2 as x vanishes...

the limit is zero

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x^2 csc = x^2/sin <--- indeterminant 0/0

L'hopitals:

2x/cos

the limit is zero