Question

(1) State the conditions for a singled-valued function of x to be continuous at x = a

(2) find the points of discontinuity of the functions
(i) x^3 + 4x + 6 / x^2 - 6x + 8

(ii) sec x

(3) Differentiate from the first princiaples (i) x^3 + 2x - 1

Answer 1

(1)
If we take the meaning of a "a singled-valued function" in the normal sense of a function (that passes the vertical line test), the conditions for continuity of f(x) at point a are:
1. f(a) exists.
2. Lim x→a+ exists and Lim x→a- exists.
3. Lim x→a+ = Lim x→a-.

(2)
If f(x) is a quotient of two polynomials, f(x) is continuous within its own domain, which excludes the vertical asymptotes. The asymptotes occur where the denominator becomes zero.
Since sec(x) = 1/cos(x), the same idea applies.

(3)
In case it is needed,
(x+h)³=x³+3x²h+3xh²+h³

Answer 2

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