(1) Prove that f(x) = 2x^{2} + 3x - 5 is continuous at all points in R. Solution
(2) Examine the continuity of the following
(i) x + sin x Solution
(ii) x^{2} cos x Solution
(iii) e^{x} tan x Solution
(iv) e^{2x} + x^{2 Solution}
(v) x ln x Solution
(vi) sin x / x^{2 Solution}
(vii) (x^{2 }- 16) / (x + 4) Solution
(viii) |x + 2| + |x - 1| Solution
(ix) |x - 2| / |x + 1| Solution
(x) cot x + tan x Solution
(3) Find the points of discontinuity of the function f, where
(ii)
(iii)
(iv)
(4) At the given point x_{0} discover whether the given function is continuous or discontinuous citing the reasons for your answer :
(5) Show that the function
is continuous on (- ∞, ∞). Solution
(6) For what value of a is this function f(x) =
continuous at x = 1 ? Solution
(7) Let f(x)
Graph the function. Show that f(x) continuous on (- ∞, ∞).
(8) If f and g are continuous functions with f(3) = 5 and lim x->3 [2 f(x) - g(x)] = 4, find g(3). Solution
(9) Find the points at which f is discontinuous. At which of these points f is continuous from the right, from the left, or neither? Sketch the graph of f.
(10) A function f is defined as follows :
Is the function continuous?
(11) Which of the following functions f has a removable discontinuity at x = x_{0}? If the discontinuity is removable, find a function g that agrees with f for x ≠ x_{0} and is continuous on R.
(i) f(x) = (x^{2} - 2x - 8)/(x + 2), x_{0} = -2 Solution
(ii) f(x) = (x^{3} + 64)/(x + 4), x_{0} = -4 Solution
(iii) f(x) = (3 - √x)/(9 - x), x_{0} = 9 Solution
(12) Find the constant b that makes g continuous on (−∞, ∞)
(13) Consider the function f (x) = x sin π/x What value must we give f(0) in order to make the function continuous everywhere? Solution
(14) The function f(x) = (x^{2} - 1) / (x^{3} - 1) is not defined at x = 1. What value must we give f(1) inorder to make f(x) continuous at x = 1 ? Solution
(15) State how continuity is destroyed at x = x_{0 }for each of the following graphs.
(i)
(ii)
(iii)
(iv)
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