How to find the asymptotes of #f(x)= 0.72^x -2#?

Question

Sadie Ackerman · Accepted Answer

#y=-2#
I have invited another mathematician to check this

Given:#" "f(x)=0.72^x-2# There are three prime conditions for this expression Condition 1: #x>0# Condition 2: #x<0# Condition 3: #x=0# '~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ #color(blue)("Consider condition 3 ")x=0# When #x=0" "f(x)# is defined as it has a definite value. Thus #color(blue)(x=0 " is NOT an asymptote")# '~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ #color(blue)("Consider condition 2 ") x<0# The #f(x)# takes the form #1/(0.72^x)-2# When #x=0": "f(x)=1-2" "# thus defined As #x# becomes bigger #0.72^2# becomes smaller so #1/(0.72^2)" "#becomes bigger Thus #lim_(xtooo) f(x)->oo# (infinity is not a number!) #color(brown)("Is this an asymptote? I have invited another mathematician to look at this bit!")# Gut feeling is that this is NOT an asymptote!!!!! '~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ #color(blue)("Consider condition 1 ") x>0# As #x# becomes increasingly larger #0.72^x# becomes increasingly smaller and smaller. So# lim_(xtooo) 0.72^x-2 -> -2color# #color(blue)(-2" Is an asymptote")#

Anna Allen · Answer

undefined To find the asymptotes of the function $ f(x) = 0.72^x - 2 $:

1. Vertical Asymptote: None, since exponential functions like $ 0.72^x $ never intersect the y-axis.

2. Horizontal Asymptote: 
   - As $ x $ approaches positive infinity, $ 0.72^x $ approaches 0 because $ 0.72 $ is between 0 and 1. Therefore, the horizontal asymptote is $ y = -2 $.