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Revision Notes on Limits, Continuity & Differentiability A. Limits Limit of a function may be a finite or an infinite number. If lim_{x→a} f(x) = ∞, it just implies that the function f(x) tends to assume extremely large positive values in the vicinity of x = a i.e. lim_{x→0} 1/|x|= ∞. A function is said to be indeterminate at any point if it acquires one of the following values at that particular point 0/0, 0 × ∞, ∞/∞, ∞-∞, 0^{0}, 1^{∞}, ∞^{0}. The 0/0 form is the standard indeterminate form. The point ‘∞’ cannot be plotted on the paper. It is just a symbol and not a number. Infinity (∞) does not obey the laws of elementary algebra. 1. ∞ + ∞ = ∞ 2. ∞ x ∞ = ∞ 3. (a/∞) = 0, if a is finite 4. (a/0) is not defined if a ≠ 0. 5. ab = 0 iff either a = 0 or b = 0 and both ‘a’ and ‘b’ are finite. In case of limits, it is important to note that the function cannot be manipulated and cancelled as in usual algebra. For example: (x^{2}-a^{2})/(x-a) = (x+a)(x-a)/((x-a)) = (x+a) This can be done in general, but in limits this is not possible until and unless (x-a) ≠ 0 or x ≠ a. The limit may exist at a point x = a even if the function is not defined at that point. If a function f is defined at a point ‘a’ i.e. f(a) exists even then it is not necessary that the limit at ‘a’ should exist. Moreover, even if the limit exists it need not be equal to f(a). Fundamental Results on Limits: Suppose lim_{x→a} f(x) = α and lim_{x→a} g(x) = β then we can define the following rules:
Limit of a function may be a finite or an infinite number.
If lim_{x→a} f(x) = ∞, it just implies that the function f(x) tends to assume extremely large positive values in the vicinity of x = a i.e. lim_{x→0} 1/|x|= ∞.
A function is said to be indeterminate at any point if it acquires one of the following values at that particular point
0/0, 0 × ∞, ∞/∞, ∞-∞, 0^{0}, 1^{∞}, ∞^{0}.
The 0/0 form is the standard indeterminate form.
The point ‘∞’ cannot be plotted on the paper. It is just a symbol and not a number.
Infinity (∞) does not obey the laws of elementary algebra.
1. ∞ + ∞ = ∞
2. ∞ x ∞ = ∞
3. (a/∞) = 0, if a is finite
4. (a/0) is not defined if a ≠ 0.
5. ab = 0 iff either a = 0 or b = 0 and both ‘a’ and ‘b’ are finite.
For example: (x^{2}-a^{2})/(x-a) = (x+a)(x-a)/((x-a)) = (x+a)
This can be done in general, but in limits this is not possible until and unless (x-a) ≠ 0 or x ≠ a.
The limit may exist at a point x = a even if the function is not defined at that point.
If a function f is defined at a point ‘a’ i.e. f(a) exists even then it is not necessary that the limit at ‘a’ should exist. Moreover, even if the limit exists it need not be equal to f(a).
Fundamental Results on Limits:
Suppose lim_{x→a} f(x) = α and lim_{x→a} g(x) = β then we can define the following rules:
lim_{x→a} k f(x) = k lim_{x→a} f(x), where k is a constant
lim_{x→a} f [g(x)] = f [lim_{x→a} g(x)] = f(m), provided f is continuous at g(x) = m.
Let f(x) be a function defined on domain D and let ‘a’ be a point such that every neighborhood of ‘a’ contains infinitely many points of D. A real number ‘l’ is called the left hand limit of f(x) at x = a iff for every ∈ > 0 there exists a δ > 0 such that a – δ < x < a ⇒ | f(x) – l| < ∈.
In such a case we write lim_{x→a-} f(x) = l.
Thus, lim_{x→a-} f(x) = l if f(x) tends to l as x tends to a from the left hand side.
Similarly, a real number ‘l’ is said to be a right hand limit of f(x) at x = a iff for every ∈ > 0, there exists a δ > 0 such that a < x < a + δ ⇒ | f(x) – l | < ∈ and we write it as lim_{x→a+} f(x) = l*.
Hence, l* is said to be the right hand limit of f(x) at x =a iff f(x) tends to l* as x tends to ‘a’ from the right hand side.
Let the function f(x) be defined in x ∈ [a, b]. At times, we might have to calculate lim _{x→b} f(x) or lim _{x→a} f(x).
In such a case, lim _{x→a} f(x) = _{x→a+} f(x) = R.H.L at x = a, as there is no left neighborhood of x = a.
Similarly, lim _{x→b} f(x) or lim _{x→b-} f(x) = L.H.L at x = b as there is no left neighborhood of x = b.
Eg: f(x) = cos^{-1} x
lim _{x→1} f(x) = lim _{x→1}^{-} cos^{-1} x = 0.
lim _{x→-1} f(x) = lim _{x→-1}^{+} cos^{-1} x = π.
The term infinite limit means that when x tends to a particular value 'a', then the limit of the function tends to infinity i.e.lim_{x→2} f(x) = ∞.
In questions which involve the evaluation of limit at infinity, the function f(x) should first be changed to g(1/x) and then we can evaluate the value at ∞.
If while calculating limits, infinite limit is encountered i.e. a zero is obtained in the denominator as x→a, then there can be two cases:
1. The term (x-a) gets cancelled from the numerator and denominator both.
2. If it does not get cancelled, then the value of the limit is put as infinity.
3. Such limits are termed as improper limits i.e. lim_{x→∞} 1/x^{2} =∞.
Let f(x), g(x) and h(x) be there real numbers having a common domain D such that h (x) ≤ f(x) ≤ g(x) ∀ x ∈ D. If lim_{x→a} h(x) = lim_{x→a} g(x)= l, then lim_{x→a} f(x) = l. This is known as Sandwich Theorem.
Let f(x) and g(x) be functions differentiable in the neighborhood of the point a, except may be at the point a itself. If lim_{x→a} f(x) = 0 = lim_{x→a} g(x) or lim_{x→a}f(x) = ∞ = lim_{x→a}g(x), then
lim_{x→a} f(x)/g(x) = lim_{x→a} f' (x)/g(x) = lim_{x→a} f' (x)/g'(x) provided that the limit on the right either exists as a finite number or is ± ∞ .
The concept of limits and continuity is quite interrelated. Limits form the base for continuity.
Let y = f(x) be a given function, and x = a is the point under consideration. Left tendency of f(x) at x = a is called its left limit and right tendency is called its right limit.
Left tendency (left limit) is denoted by f(a - 0) or f(a^{-}) and right tendency (right limit) is denoted by f(a + 0) or f(a+) and are written as
where 'h' is a small positive number.
Thus for the existence of the limit of f(x) at x = a, it is necessary and sufficient that f(a – 0) = f(a + 0), if these are finite or f(a-0) and f(a+0) both should be either +∞ or -∞.
It is quite possible for a function to have infinite limit as x → a. If lim_{x→a} f(x) = ∞, it would simply mean that function has tendency to assume very large positive values in the neighborhood of x = a, for example lim_{x→0} 1/|x| = ∞.
Some of the methods of evaluating limits include:
Consider the following limits (i) lim_{x→a} f(x) and (ii) lim_{x→a} φ(x)/Ψ(x).
If f(a) and φ(a)/Ψ(a) exist and are fixed real numbers and Ψ(a) ≠ 0, then we say that lim_{x→a} f(x) = f(a) and lim_{x→a} φ(x)/Ψ(x) = φ(a)/Ψ(a).
Consider lim_{x→a} f(x)/g(x). If by substituting x = a, f(x)/g(x) reduces to the form 0/0, then (x-a) is a factor of both f(x) and g(x). So, we first factorize f(x) and g(x) and then cancel out the common factor to evaluate the limit.
This method is generally used in cases where either the numerator or the denominator or both involve expression consists of square roots and on substituting the value of x, the rational expression takes the form 0/0, ∞/∞.
We know that lim x→∞ 1/x = 0 and lim x→∞ 1/x^{2} = 0.
Then lim _{x→∞} f(x) = lim _{y→0} f(1/y).
When infinite limit is encountered i.e. '0' in the denominator as x→a, then either (x-a) get cancelled from numerator and denominator both and if it is not possible then the limit's value is put as infinity. Such limits are called improper limits i.e. lim_{x→∞} 1/x^{2} = ∞.
Let f(x), g(x) and h(x) be three real numbers having a common domain D such that h(x) ≤ f(x) ≤ g(x) ∀ x ∈ D. If lim_{x→a} h(x) = lim_{x→a} g(x) = l, then lim_{x→a} f(x) = l. This is known as Sandwich Theorem.
The L Hospital’s rule is used in cases where the limit obtained is in the form of 0/0, ∞/∞, ∞ – ∞,0, ∞, 0^{0}, ∞^{0}, 1^{∞}. These are known as indeterminate forms. According to this rule,
Let f(x) and g(x) be functions differentiable in the neighborhood of the point a, except may be at the point a itself. If lim _{x→a} f(x) = 0 = lim_{x→a} g(x) or lim_{x→a}f(x) = ∞ = lim_{x→a} g(x), then
lim_{x→a} f(x)/g(x) = lim_{x→a} f ’(x)/g'(x) provided that the limit on the right either exists as a finite number or is ± ∞ .
A function f(x) is said to be continuous at x= a if
lim_{x→a}^{-} f(x)= lim_{x→a}^{+} f(x)=f(a)
Thus, unlike limits, for continuity it is essential for the function to be defined at that particular point and the limiting value of the function should be equal to f(a).
The function f(x) will be discontinuous at x = a in either of the following situations:
lim_{x→a}^{-} f(x) and lim_{x→a}^{+} f(x) exist but are not equal.
lim_{x→a}^{-} f(x) and lim_{x→a}^{+} f(x) exist and are equal but not equal to f(a).
f(a) is not defined.
At least one of the limits does not exist.
If you are required to comment on the continuity of a function, then you may just look for the points on the domain where the function is not defined.
If the functions f(x) and g(x) are both continuous at x = a then the following results hold true:
1. cf (x) is continuous at x = a where c is any constant.
2. f(x) + g(x) is continuous at x = a.
3. f(x).g(x) is continuous at x= a.
4. f(x)/g(x) is continuous at x= a, provided g(a) ≠ 0.
If a function f is continuous in (a, b), it means it is continuous at every point of (a, b).
If f is continuous in [a, b] then in addition to being continuous ay every point of domain, f should also be continuous at the end points i.e. f(x) is said to be continuous in the closed interval [a, b] if
1. f(x) is continuous in (a, b)
2. lim_{x→a}^{+} f(x) = f(a)
3. lim_{x→a}^{-} f(x) = f(a)
While solving problems on continuity, one need not calculate continuity at every point, in fact the elementary knowledge of the function should be used to search the points of discontinuity.
In questions like this where a function h is defined as
h(x) = f(x) for a < x < b
g(x) for b < x < c
The functions f and g are continuous in their respective intervals, then the continuity of function h should be checked only at the point x = b as this is the only possible point of discontinuity.
If the point ‘a’ is finite, then the necessary and sufficient condition for the function f to be continuous at a is that lim_{x→a}^{-} f(x) and lim_{x→a}^{+} f(x) should exist and be equal to f(a).
A function continuous on a closed interval [a, b] is necessarily bounded if both a and b are finite. This is not true in case of open interval.
If the function u = f(x) is continuous at the point x = a, and the function y = g(u) is continuous at the point u = f(a), then the composite function y = (gof)(x) = g(f(x)) is continuous at the point x = a.
Given below is the table of some common functions along with the intervals in which they are continuous:
Functions f(x)
Interval in which f(x) is continuous
Constant C
(-∞,∞)
b^{n}, n is an integer > 0
|x-a|
x^{-n}, n is a positive integer.
(-∞,∞) – {0}
a_{0}x^{n} + a_{1}x^{n-1 }+........ + a_{n-1}x + a_{n}
p(x)/q(x), p(x) and q(x) are polynomials in x
R - {x:q(x) = 0}
sin x
R
cos x
tan x
R-{nπ:n = 0,±1,........}
cot x
R-{(2n-1)π/2: n = 0,±1,± 2,........}
sec x
e^{x}
ln x
(0, ∞)
If lim_{x→a}^{-} f(x) = L_{1} and lim_{x→a}^{+} f(x) = L_{2}, where L_{1 }and L_{2 }are both finite numbers then it is called discontinuity of first kind or ordinary discontinuity.
A function is said to have discontinuity of second kind if neither lim_{x→a}^{+} f(x) nor lim_{x→a}^{-} f(x) exist.
If any one of lim_{x→a}^{+} f(x) or lim_{x→a}^{-} f(x) exists and the other does not then the function f is said to have mixed discontinuity.
If lim_{x→a} f(x) exists but is not equal to f(a), then f(x) has removable discontinuity at x = a and it can be removed by redefining f(x) at x = a.
If lim_{x→a} f(x) does not exist, then we can remove this discontinuity so that it becomes a non-removable or essential discontinuity.
A function f(x) is said to have a jump discontinuity at a point x = a if, lim_{x→a}^{- }f(x) ≠ lim_{x→a}^{+} f(x) and f(x) and may be equal to either of previous limits.
The concepts of limit and continuity are closely related. Whether a function is continuous or not can be determined by the limit of the function.
A function f(x) is said to be differentiable at point P iff there exists a unique tangent at point P.
A function f(x) is differentiable at a point P iff the curve does not have P as a corner point i .e. the curve is not differentiable at the points where there are holes or sharp edges.
The function f(x) is said to be differentiable in interval (a, b) if f(x) is differentiable at every point of interval (a, b).
The function f(x) is said to be differentiable in closed interval [a, b] if in addition to being differentiable in (a, b), f(x) is differentiable at x = a from the right and at x = b from the left.
A function which is differentiable at x = a is necessarily continuous at x = a but not vice versa.
The sum, difference, product and quotient of two differentiable functions is differentiable.
The composition of differentiable functions is a differentiable function.
Continuity and differentiability of functions:
The identity, exponential and logarithmic functions are continuous and differentiable in their domain.
The greatest integer function, least integer function and the fractional part functions are continuous and differentiable at all points except at integral values.
The signum function is continuous and differentiable at all points except at x = 0.
The polynomial function is continuous and differentiable everywhere.
The trigonometric functions i.e. sine, cosine, tangent, cotangent, secant and cosecant are continuous and differentiable in their domain.
The inverse trigonometric functions are continuous and differentiable in their domain.
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