Limits
One Shot
Nishant Vora
B.Tech - IIT Patna
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Concept of
Limits
Concept of Limits
f(x) =
Since f(0) is not defined, so we will try to find the value of f(x)
at x = 0+ and x = 0﹣
x⟶a x=a
Understanding Limits from Graph
f(x) =
Existence of Limit
If LHL = RHL = finite then limit exist
Graphical
Understanding
Understanding Limits from Graph
A. 0
B.
C.
D. Does not exists
Indeterminate
Forms
7 Indeterminate Forms
0/0 and ∞/∞
Form
Methods for Solving
1. Factorisation Method
2. Rationalisation Method
3. L’Hopital Rule
Evaluate
Factorisation Method L’Hopital Rule
Evaluate
Factorisation Method L’Hopital Rule
2. Rationalisation Method
Evaluate
3. L’Hopital Method
Evaluate
Let f : R ⟶ R be a function such that f(2) = 4 and f’ (2) = 1.
Then, the value of
(JEE Main 2021)
A. 4
B. 8
C. 16
D. 12
Let f(x) = x6 + 2x4 + x3 + 2x + 3, x ∈ R. Then the natural number n for
which
(JEE Main 2021)
If
then the value of n is equal to _.
(JEE Main 2020)
Basic
Questions
[JEE Main 2020]
The value of limit
is _______
[JEE Adv 2020]
and
and
Evaluate Shortcut
🔥NVStyle
If deg(Nr) > deg(Dr) If deg(Nr) = deg(Dr) If deg(Nr) < deg(Dr)
0 x ∞ and ∞-∞
Form
Method to solve ∞-∞, 0x∞
S-1 Convert to or
S-2 Use factorization/ rationalization/ L’Hopital
Evaluate
(JEE Main 2021)
A. (1, 1/2)
B. (1, -1/2)
C. (-1, 1/2)
D. (-1, -1/2)
Reverse
Questions
Reverse Question
If Find a and b.
(JEE Main 2021)
A. -4
B. 5
(JEE Main 2019)
C. -7
D. 1
Binomial
Approximation
Binomial Approximation
Let L = if L is finite, then
A. a=2
B. a=1
C. L = 1/64 [JEE Adv 2009]
D. L = 1/32
Super Table
🔥 Super Table
f(x) g(x) f±g f.g and f/g
Exist Exist Exist Exist
Exist D.N.E D.N.E May Exist
D.N.E D.N.E May Exist May Exist
Let α (a) and β (a) be the roots of the equation
where a > -1. Then
[JEE 2012]
A.
B.
C.
D.
Squeeze (or
Sandwich)
Theorem
If f(x) ≤ g(x) ≤ h(x)
c
If 4x - 9 ≤ f(x) ≤ x2 - 4x + 7 ∀ x ≥ 0 find
A. -7
B. 7
C. 1/7
D. D.N.E.
2x ≤ g(x) ≤ x4 - x2 + 2 for all x then
A. 2
B. -2
C. 0
D. D.N.E.
The value of where r is non-zero real numbers
and [r] denotes the greatest integer less than or equal to r, is equal to :
A. r/2 (JEE Main 2021)
B. r
C. 2r
D. 0
Standard
Forms
Remember
Remember
(JEE Main 2021)
A. π2
B. 2π2
C. 4π2
D. 4π
The value of the limit
(JEE Main 2021)
A. -1/2
B. -1/4
C. 0
D. 1/4
(JEE Main 2019)
A. 4√2
B. √2
C. 2√2
D. 4
A. 0
B. 2
(JEE Main 2019)
C. 4
D. 1
then the value of k is_
(JEE Main 2020)
(JEE Main 2020)
A. is equal to √e
B. is equal to 1
C. is equal to 0
D. Does not exist
Expansions
Maclaurin series
Expansions
Expansions
Expansions
A. 1/6
(JEE Main 2021)
B. 1/2
C. 6
D. 2
Evaluate the following limit
(JEE Main 2021)
(JEE Main 2021)
Expansions
Let e denote the base of natural logarithm. The value of real number a for
which the right hand limit
is equal to a nonzero real number, is _________ [JEE Adv 2020]
00 and ∞0 Form
Method: Take log both sides
Evaluate
1∞ Form
Practical Proof
Evaluate
A. 1
B. e
C. e2
D. e-1
(JEE Main 2021)
A. (1, -3)
B. (-1, 3)
C. (-1, -3)
D. (1, 3)
(JEE Main 2020)
A. e
B. 2
C. 1
D. e2
One sided
Limits
One Sided Limits
Let then log p is equal to -
A. 2
B. 1 [Main 2016]
C. 1/2
D. 1/4
(JEE Main 2021)
A. π
B. 0
C. π4
D. π/2
Limits Involving
G.I.F.
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