what the questions want

Question 1

checking whether a sequence is null:

Answer 1

find some n such that the sequence would converge to n, think about the list of standard null sequences (they often have a couple fractions in a brackets all to the power of n, for that find some n such that the total of the fractions <1 which would then converge) - if there isn’t a way to do that, just write out why it doesn’t work

Question 2

computing a limit:

Answer 2

literally just jangle it around sorry, if it’s a horrible square root thing expand it to a fraction, just make it equal a number and voila limit, for trig make use of the fact that cos and sin range between 0 and 1 so you can place it between 2 values, try and make those null ideally, and use the sandwich rule

Question 3

is x series convergent/divergent:

Answer 3

check if the sequence creating it is null

Question 4

show that if the series of an converges, an is null:

Answer 4

sn -> r as n -> infinity, an = s(n) - s(n-1), so an -> r-r = 0 as n -> infinity

Question 5

compute the derivative:

Answer 5

do it from first principles especially if there’s a modulus involved, check the product and chain rule and stuff after

Question 6

prove smth using the intermediate value theorem:

Answer 6

set g(x) = f(x) - (whatever f(x) was set as equal to), check 0 for x and check the result is still in the set range, check 1 the same, then just say you’re applying the ivt so there must be a value equal to it cause there must be some x that sets g(x)=0
this may need a h(x) to get a y(x)-(z) but that’s okay

Question 7

prove that if f and g are continuous, that (some combo of f and g) are also continuous:

Answer 7

write out the diff by first principles of the combo, then jangle it until you get the diff by first principles individually

Question 8

convergence of series and algebra of limits:

Answer 8

use the comparison test, try and find a way to make the series of bn equal to L*the series of an where L is the upper bound of the given convergent sequence function (expand bn with ans to match the given sequence function, then replace that with L with the aim of removing bns)

Question 9

suppose a continuous function that isn’t constantly c, show that there’s a nonempty interval (a,b) such that f(x)!=c for all x in (a,b):

Answer 9

f isn’t constantly c so there must be some z in (a,b) such that f(z)!=c
f is continuous so as lim(x->z), f(x)=f(z)
take epsilon < |f(z)|
then by the definition of a limit there must be an open interval (z-delta, z+delta) such that |f(z)-f(x)|<epsilon for all x in that interval{z}
therefore by choice of epsilon, f(x)!=c for all x in (z-delta, z+delta)

Question 10

prove that if an is convergent then the limit is unique:

Answer 10

assume two different limits L and L’ that aren’t equal to each other, choose 0<epsilon<|(L-L’)/3|
then there exists N such that |an-L|<epsilon and |an-L’|<epsilon>N
but then for n>N, |L-L'|=|L-an+an-L'|<=|an-L|+|an-L'|<2epsilon<(2/3)|L-L'| which is a contradiction, so L=L'</epsilon>