Certainly! Let's analyze the long run behavior of each of the given functions step by step.
### Part (a)
Function: [tex]$14(0.9)^x$[/tex]
As [tex]\( x \)[/tex] tends to infinity ([tex]\( x \to \infty \)[/tex]):
- The base of the exponent is [tex]\( 0.9 \)[/tex], which is less than 1.
- When the base of an exponential function is a number between 0 and 1, the function [tex]\( (0.9)^x \)[/tex] tends to 0 as [tex]\( x \)[/tex] increases indefinitely.
Thus, multiplying [tex]\( 14 \)[/tex] by a value that approaches 0 will result in:
[tex]\[ 14(0.9)^x \longrightarrow 0 \][/tex]
### Part (b)
Function: [tex]\( 7(2.3)^t \)[/tex]
As [tex]\( t \)[/tex] tends to negative infinity ([tex]\( t \to -\infty \)[/tex]):
- The base of the exponent is [tex]\( 2.3 \)[/tex], which is greater than 1.
- When the base of the exponential function is greater than 1 and the exponent tends to negative infinity, the function [tex]\( (2.3)^t \)[/tex] approaches 0 because any positive number raised to a significantly large negative power becomes a very small positive number.
Thus, multiplying [tex]\( 7 \)[/tex] by a value that approaches 0 will result in:
[tex]\[ 7(2.3)^t \longrightarrow 0 \][/tex]
### Part (c)
Function: [tex]\( 0.8 \left( 3 - (0.4)^t \right) \)[/tex]
As [tex]\( t \)[/tex] tends to infinity ([tex]\( t \to \infty \)[/tex]):
- The base of the exponent is [tex]\( 0.4 \)[/tex], which is less than 1.
- When the base of an exponential function is a number between 0 and 1, the function [tex]\( (0.4)^t \)[/tex] tends to 0 as [tex]\( t \)[/tex] increases indefinitely.
- As [tex]\( t \to \infty \)[/tex], [tex]\( (0.4)^t \to 0 \)[/tex], so the expression inside the parentheses tends to [tex]\( 3 - 0 \)[/tex] which is 3.
Thus:
[tex]\[ 0.8 \left( 3 - (0.4)^t \right) \longrightarrow 0.8 \times 3 = 2.4 \][/tex]
### Conclusion
Combining all parts, the long run behaviors of each function are:
(a) [tex]\[ 14(0.9)^x \longrightarrow 0 \][/tex]
(b) [tex]\[ 7(2.3)^t \longrightarrow 0 \][/tex]
(c) [tex]\[ 0.8 \left( 3 - (0.4)^t \right) \longrightarrow 2.4 \][/tex]
