How to find the asymptotes of #f(x) = 6^(x-2)#?

Find the limit in either infinite direction.

Note that this is just a mental process to undergo.

We can check a graph:

graph{6^(x-2) [-8.535, 13.965, -2.52, 8.73]}

To find the asymptotes of the function ( f(x) = 6^{x-2} ), follow these steps:

1. Determine vertical asymptotes by identifying any values of ( x ) that would make the function undefined. In this case, since ( 6^{x-2} ) is defined for all real numbers, there are no vertical asymptotes.

2. Find horizontal asymptotes by analyzing the behavior of the function as ( x ) approaches positive or negative infinity. As ( x ) approaches positive infinity, ( 6^{x-2} ) increases without bound, so there is no horizontal asymptote in that direction. Similarly, as ( x ) approaches negative infinity, ( 6^{x-2} ) approaches zero, indicating a horizontal asymptote at ( y = 0 ).

Therefore, the horizontal asymptote of the function ( f(x) = 6^{x-2} ) is ( y = 0 ).