Limits

Question 1

What are the techniques that can be used to solve a complex limit?

Answer 1

We can use:
Logarithmic Rules,
Binomial Theorem
L’hopitals rule
Simple evaluation
Squeeze rule
Rearranging variables for 0 instead of infinity or vice versa
(We often use a combination of these rules)

Question 2

Points on using ln rules:

Answer 2

We use when we have an exponent that makes the evaluation difficult.
Set L = to out limit
Take ln of both sides inside the limit.
Rearrange to make a quotient (a/b)
Then will often use l’hopitals rule, then rearrange for L.

Question 3

d/dx(ln(u))

Answer 3

U’/U

Question 4

Simplified binomial theorem equation and conditions

Answer 4

(1+x)^p =1 + px+ (p(p-1)x^2)/2! + (p(p-1)(p-2)x^3/3! …..
valid for |p|<1 and x>0

Question 5

Process for simplified binomial

Answer 5

Often used when there is a square root in the limit or similar, we can often expand and take out a factor or show things will equal 0.

Question 6

L’hopitals rule equation

Answer 6

For our limit: if a/b = 0/0 or Inf/Inf then our limit will equal the d/dx(a)/d(dx(b)

Question 7

Squeeze rule

Answer 7

If we can set up an interval where g(x)<=f(x)<=h(x) and we know both g(x) and h(x) (they equal one another then we can show that f(x) will equal that.

Question 8

Setup for squeeze rule

Answer 8

If triganometric function involved then use our knowledge of its range to set up an interval, then we manipulate our function to make it look like our original limit and then solve.

Question 9

Technique to simplify logarithms in questions

Answer 9

If handy to flip fraction inside logarithm then we can put it to a negative power