Math Help

Examples involving Continuity

Example 1: f(x) = 1/(x + 1) for x > ˗ 1 Suppose a > ˗ 1

lim f(x) = lim [1/(x+1)] = 1/(a+1) = f(a)

x → a x → a

Example 2: f(x) = 1/( x² + 1) Let a be any finite value of x

lim f(x) = lim [1/( x² + 1)] = 1/( a² + 1) = f(a)

x → a x → a

So f(x) is continuous for (-∞, ∞) .

Example 3: f(x) = ( x – 7 ) / | x – 7 |

Now lim f(x) does not exist since lim f(x) ≠ lim f(x)

x → 7 x → 7^˗ x → 7⁺

So f(x) is discontinuous at x = 7.

But the limit exists provided that either x < 7 or x > 7.

i.e. x = 6 lim f(x) = lim (6 ˗ 7) / |6 – 7| = ˗ 1

x → 6 x → 6

i.e. x = 8 lim f(x) = lim (8 ˗ 7) / |8 – 7| = + 1

x → 8 x → 8

So f(x) is continuous for any x where (-∞ , 7), (7 , ∞)

Squeeze Law Example: g(x) = sin (x) , f(x) = -x, h(x) = x

Note: | sin x | ≤ 1 so - x ≤ sin x ≤ x for x ≠ 0 and near x = 0. Then

lim f(x) = 0 = lim h(x)

x → 0 x → 0

then lim[ sin(x)] = lim g(x) = 0

x → 0 x → 0