Three requirements have to be satisfied for the continuity of a function y = f(x) at x = x0 :
(i) f(x) must be defined in a neighbourhood of x0 (i.e., f(x0) exists);
(ii) lim x->x0 f(x) exists.
(iii) f(x0) = lim x -> x0 f(x)
To know the points to be remembered in order to decide whether the function is continuous at particular point or not, you may look into the page " How to check continuity of a function, if interval is not given "
Question 1 :
Examine the continuity of the following
x2 cos x
Solution :
Let f(x) = x2 cos x
(i) From the given function, we know that both "x" and "cos x" are defined for all real numbers.
(ii) lim x-> x0 f(x) = lim x-> x0 x2 cos x
By applying the limit, we get
= x02 cos x0 -------(1)
(iii) f(x0) = x02 cos x0 -------(2)
From (1) and (2)
lim x-> x0 f(x) = f(x0)
Hence the given function is continuous for all real numbers.
Question 2 :
Examine the continuity of the following
ex tan x
Solution :
Let f(x) = ex tan x
From the given function, we know that the exponential function is defined for all real values.But tan is not defined at π/2.
So, the function is continuous for all real values except (2n+1)π/2.
Hence the answer is continuous for all x ∈ R- (2n+1)π/2.
Question 3 :
Examine the continuity of the following
e2x + x2
Solution :
Let f(x) = e2x + x2
(i) From the given function, we know that both "x" and "exponential function" are defined for all real numbers.
(ii) lim x-> x0 f(x) = lim x-> x0 e2x + x2
By applying the limit, we get
= e2x0 + (x0)2 -------(1)
(iii) f(x0) = e2x0 + (x0)2 -------(2)
From (1) and (2)
lim x-> x0 f(x) = f(x0)
Hence the given function is continuous for all real numbers.
Question 4 :
Examine the continuity of the following
x ln x
Solution :
Let f(x) = x ln x
(i) From the given function, we know that both "x" defined for all real values, but logarithmic function is defined only on (0, ∞)
Let us take x0 from (0, ∞)
(ii) lim x-> x0 f(x) = lim x-> x0 x ln x
By applying the limit, we get
= x0 ln x0 -------(1)
(iii) f(x0) = x0 ln x0 -------(2)
From (1) and (2)
lim x-> x0 f(x) = f(x0)
Hence the given function is continuous on (0, ∞).
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