How do you find vertical, horizontal and oblique asymptotes for #g(x)=5^x#?

Question

Ariana Alvarez · Accepted Answer

There will be a horizontal asymptote at #y = 0#.

The domain of any exponential function is #{x| x in RR}#, so no vertical asymptotes. As for horizontal asymptotes, the following limit will be determinative: #lim_(x-> -oo) 5^x# Calling the limit #L#, we have: #L = lim_(x->-oo) 5^x# If we think about it, we realize that the closer the number #x = a# gets to #-oo#, the closer #L# will get to #0#. This is because #a^-n = 1/a^n#, and #1/oo = 0#. So something like #5^-1000# would be very close to #0#. Therefore, we can say that #g(x)# has a horizontal asymptote at #y = 0#. There will be no oblique asymptote for #g(x)#. Hopefully this helps!

Sophie Adams · Answer

undefined To find the vertical, horizontal, and oblique asymptotes for the function $ g(x) = 5^x $:

1. Vertical asymptote: There are no vertical asymptotes for exponential functions like $ 5^x $.

2. Horizontal asymptote: As $ x $ approaches negative or positive infinity, $ 5^x $ grows without bound. Therefore, there is no horizontal asymptote.

3. Oblique asymptote: There is no oblique asymptote for the function $ g(x) = 5^x $. Oblique asymptotes typically occur in rational functions when the degree of the numerator is one greater than the degree of the denominator. Since $ 5^x $ is an exponential function, it does not have an oblique asymptote.