limit

Question

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

Answer 1

The function is f(x) = 1/(x - 10).

The table lists the value of f(x) for several x-values approaches 10 from the left.

x	f(x) = 1/(x-10)	[x, f(x)]
9.5	f(x) = 1/(9.5 - 10) = 1/(-0.5) = - 2	(9.5, -2)
9.9	f(x) = 1/(9.9 - 10) = 1/(-0.1) = - 10	(9.9, - 10)
9.99	f(x) = 1/(9.99 - 10) = 1/(-0.01) = - 100	(9.99, -100)
9.999	f(x) = 1/(9.999 - 10) = 1/(-0.001) = - 1000	(9.999, -1000)
9.9999	f(x) = 1/(9.9999 - 10) = 1/(-0.0001) = - 10000	(9.9999, -10000)
9.99999	f(x) = 1/(9.99999 - 10) = 1/(-0.00001) = - 100000	(9.99999, -100000)
10	f(x) = 1/(10 - 10) = - 1/0 = - ∞	(10, -∞)

 

The function f(x) = 1/(x-10) graph is

 

Observe the graph and table,

when x approaches 10 from the left, (x - 10) is a small negative number. Thus, the quotient 1/(x-10) is a large negative number and f(x) approaches negative infinity from left side of x = 10. So, we can conclude that x = 10 is a vertical asymptote of the graph of f(x) and