How do you find the asymptotes for #4^(x-5)-5#?

Question

Sadie Alexander · Accepted Answer

See below.

Vertical asymptotes occur where the function is undefined. This particular function is defined for all x, so no vertical asymptotes. To find any horizontal asymptotes will need to examine the end behaviour as #x-> +-oo# as #x->oo# , #color(white)(88)4^(x-5)-5->oo# For #x<5#: #4^(x-5)-5# becomes #1/4^(x-5)-5# as #x->oo# ,#color(white)(88)1/4^(x-5)-5->-5# ( we used positive infinity here, since we rewrote the expression ) So the horizontal asymptote is the line: #y= -5# graph{4^(x-5)-5 [-16.02, 16.01, -8.01, 8.01]}

Caleb Alexander · Answer

undefined To find the asymptotes for $4^{x-5} - 5$:

1. Examine the function $4^{x-5}$:
   - There are no vertical asymptotes because exponential functions do not have vertical asymptotes.
   - Determine if there are horizontal asymptotes by analyzing the behavior as $x$ approaches positive or negative infinity.

2. As $x$ approaches positive infinity:
   - $4^{x-5}$ grows without bound, meaning it approaches infinity.
   - Thus, there is no horizontal asymptote at positive infinity.

3. As $x$ approaches negative infinity:
   - $4^{x-5}$ approaches zero since the base is greater than 1, and the exponent tends to negative infinity.
   - Therefore, the horizontal asymptote is at $y = -5$.

4. The horizontal asymptote is $y = -5$. There are no vertical asymptotes.