Limits and Continuity

Question 1

1^2 + 2^2 + 3^2 + 4^2 +….+ n^2 = ?

Answer 1

n(n+1)(n+6)/3

Question 2

How to solve an ∞ - ∞ question in limits?

Answer 2

Rationalise or double rationalise it.

Question 3

lim
x→π/2 [1+tan 2x][(1−sinx)
(1−tan 2x)π−2x]^3

Answer 3

Answer: 1/32
substitute x->pi/2 + h

Question 4

∞
∏. (1-1/n^2) evaluate
n=2

Answer 4

Answer: 1/2
expand in (13)(23)…..
2^2 * 3^3 *…. form

Question 5

Maclouren’s series
sin x

Answer 5

x- x^3/3! + x^5/5! - x^7/7! +…

Question 6

Maclouren’s series
cos x

Answer 6

x - x^2/2! + x^4/4! - x^6/6! +…

Question 7

Maclouren’s series
tan x

Answer 7

x + x^3/3! + 2x^5/15 +17x^7/315 +….

Question 8

Maclouren’s series
normal formula:

Answer 8

f(x) = f(0) + f’(0)x + f’‘(0)x^2/2! + f’’‘(0)x^3/3! + …

Question 9

how do you solve a question in 0/0 form?

Answer 9

Use L’Hopital’s rule

Question 10

How to solve x -> ∞

Answer 10

->degree of numerator = degree of denominator
->Put x=1/y and apply x->0

Question 11

when are limits undefined?

Answer 11

-> LHL not equal to RHL
-> Infinte oscillations
-> towards infinity

Question 12

How to solve 1^∞ form?

Answer 12

e^lim x->a [f(x)-1]g(x)

Question 13

What is jump of discontinuity?

Answer 13

RHL-LHL

Question 14

What are the 6 points of suspicion?

Answer 14

1. When denominator gets 0
2. When GIF gets Z
3. When { } gets Z
4. sgm wherever gets 0
5. Wherever function changes its definition
6. Extreme points

Question 15

Maclouren’s series
(1+x)^n

Answer 15

1 + nx + n(n-1)x^2/2! + …

Question 16

What is Rolle’s theorem?

Answer 16

1. Continuous in [a,b]
2. Differentiable in (a,b)
   3.f(a)=f(b)
3. Then f’(c) = 0

Question 17

Maclouren’s series
e^x

Answer 17

1+ x^2/2! + x^3/3! + …

Question 18

lim x-> 0 1-cosx/x^2

Answer 18

1/2

Question 19

1 + 2 + 3 + …

Answer 19

n(n+1)/2