First slide

Continuity

Question

$f (x) = \lim_{n \to \infty} \frac{{(x - 1)}^{2 n} - 1}{{(x - 1)}^{2 n} + 1}$ is discontinuous at

Moderate

A

x = 0 only
B

x = 2 only
C

x = 0 and 2
D

x = 0, 1,2 only

Solution

f(x) is discontinuous when x = 0,2 $f (x) = \lim_{n \to \infty} \frac{1 - \frac{1}{{[{(x - 1)}^{2}]}^{n}}}{1 + \frac{1}{{[{(x - 1)}^{2}]}^{n}}}$ $= \{\begin{matrix} 1, & x < 0 \\ 0, & x = 0 \\ - 1, & 0 < x < 2 \\ 0, & x = 2 \\ 1, & x > 2 \end{matrix}$
Thus, f(x) is discontinuous at x = 0,2.

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