Continuity at a Point 3

Question

Use graphs and tables to find the limit and identify any vertical asymptotes of .

Answer 1

The function is f(x) = 1/(x-6).

The table lists the value of f(x) for several x-values approaches 6 from the left.

x	f(x) = 1/(x-6)	[x, f(x)]
5.5	f(x) = 1/(5.5 - 6) = 1/(-0.5) = - 2	(5.5, -2)
5.9	f(x) = 1/(5.9 - 6) = 1/(-0.1) = - 10	(5.9, - 10)
5.99	f(x) = 1/(5.99 - 6) = 1/(-0.01) = - 100	(5.99, -100)
5.999	f(x) = 1/(5.999 - 6) = 1/(-0.001) = - 1000	(5.999, -1000)
5.9999	f(x) = 1/(5.9999 - 6) = 1/(-0.0001) = - 10000	(5.9999, -10000)
5.99999	f(x) = 1/(5.99999 - 6) = 1/(-0.00001) = - 100000	(5.99999, -100000)
6	f(x) = 1/(6 - 6) = - 1/0 = - ∞	(6, -∞)

 

The function f(x) = 1/(x-6) graph is

 

Observe the graph and table,

when x approaches 6 from the left, (x - 6) is a small negative number. Thus, the quotient 1/(x-6) is a large negative number and f(x) approaches negative infinity from left side of x = 6. So, we can conclude that x = 6 is a vertical asymptote of the graph of f(x) and