limits

Question

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

Answer 1

The function is f(x) = 1/(x-9).

The table lists the value of f(x) for several x-values approaches 9 from the left.

x	f(x) = 1/(x-9)	[x, f(x)]
8.5	f(x) = 1/(8.5 - 9) = - 1/0.5 = - 2	(8.5, -2)
8.9	f(x) = 1/(8.9 - 9) = - 1/0.1 = - 10	(8.9, - 10)
8.99	f(x) = 1/(8.99 - 9) = - 1/0.01 = - 100	(8.99, -100)
8.999	f(x) = 1/(8.999 - 9) = - 1/0.001 = - 1000	(8.999, -1000)
8.9999	f(x) = 1/(8.9999 - 9) = - 1/0.0001 = - 10000	(8.9999, -10000)
8.99999	f(x) = 1/(8.99999 - 9) = - 1/0.00001 = - 100000	(8.99999, -100000)
9	f(x) = 1/(9 - 9) = - 1/0 = - ∞	(9, -∞)

 

The function f(x) = 1/(x-9) graph is

 

Observe the graph and table,

when x approaches 9 from the left, (x - 9) is a small negative number. Thus, the quotient 1/(x-9) is a large negative number and f(x) approaches negative infinity from left side of x = 9. So, we can conclude that x = 9 is a vertical asymptote of the graph of f(x) and

.