SINGLE VARIABLE
CALCULUS
19MAT101
LIMITS
Section 2.1
Exercise 2.1 - Problems 1 to 10
Limits
Limit of a function:
The function f(x) is said to have limit L at a point x = a if the
function value approaches ‘L’ as ‘x’ approaches ‘a’. We write
this as lim 𝑓 𝑥 = 𝐿
𝑥→𝑎
Since x approaches a from the left and from the right we
have the notion of left hand limit and the right hand limit.
The left hand limit is denoted by lim− 𝑓(𝑥) and the right
𝑥→𝑎
hand limit is denoted by lim+ 𝑓(𝑥).
𝑥→𝑎
The limit exists only if the left hand limit and right hand limit
exists and are equal.
From the given function find the limits at the points
x = -1, 0, 1, 2, 3,4.
CONTINUITY
Section 2.6
Exercise Problem 1 to 49
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Dr. Geetha K N
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Continuity
A function f(x) is said to be continuous at an interior point
x = a, if
(i)
f is defined at a
(ii) lim 𝑓 𝑥 exists
𝑥→𝑎
(iii) lim 𝑓(𝑥) = f(a)
𝑥→𝑎
A function is said to be continuous at an end point ‘a’, if ‘f’ is
defined at ‘a’
If the left hand limit exists at the right end point ’a’ and equal to
f(a) then the function is said to be left continuous at the point ‘a’
If the right hand limit exists at the left end point ’a’ and equal to
f(a) then the function is said to be right continuous at the point ‘a’
If the function is continuous at end points then left end point is
right continuous and right end point is left continuous.
If the function is not continuous at a point then we call it as
discontinuous.
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Dr. Geetha K N
13
Types of Discontinuity
Discontinuity
Discontinuity when
When limit exists
limit does not exists
LHL = RHL ≠ f(x)
LHL ≠ RHL
Removable
Jump
Infinite
Oscillating
Discontinuity
(limit is
(limit is
(not fixed)
finite)
infinite)
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Dr. Geetha K N
15
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Dr. Geetha K N
18
Find the continuous extension function at x = 0
Geometrically a function is continuous if the curve is smooth
Types of Non-differentiability
A function fails to have derivatives at points where its curve fails to be
smooth such as points where the graph has
1)
a discontinuity at x = a
2)
Corner Point when LHD ≠ RHD, but both are finite
3)
Cusp: when LHD = ±∞ and RHD = ∓∞ (we get a common tangent)
4)
Vertical tangent: When LHD =RHD = +∞
or -∞.
Check whether the following functions have derivative at the
given point. If it has a derivative, find the derivative at the point.
If not, find out the type of non-differentiability.
1. 𝑦 = 𝑥 , 𝑥 = 0
2
𝑥
𝑥 < 0 at x = 0
2. 𝑦 =
𝑥 𝑥 ≥0
3. f(x) = x3 + x, at x = 0
The only points where the can have a extreme value (local or global)
are: