The y-intercept of the function f(x) is (0, 10/3)
The limits of the function
The function is given as:
[tex]f\left(x\right)=\frac{30}{1+2^{3-x}}[/tex]
As x approaches infinity, we have:
[tex]\lim_{x \to \infty} \frac{30}{1+2^{3-x}}[/tex]
This gives
[tex]\lim_{x \to \infty} \frac{30}{1+2^{3-\infty}}[/tex]
[tex]\lim_{x \to \infty} \frac{30}{1+2^{-\infty}}[/tex]
Evaluate
[tex]\lim_{x \to \infty} \frac{30}{1}[/tex]
[tex]\lim_{x \to \infty} 30[/tex]
As x approaches negative infinity, we have:
[tex]\lim_{x \to -\infty} \frac{30}{1+2^{3+x}}[/tex]
This gives
[tex]\lim_{x \to -\infty} \frac{30}{\infty}[/tex]
Evaluate
[tex]\lim_{x \to -\infty} 0[/tex]
Hence, the limits of the function are [tex]\lim_{x \to \infty} 30[/tex] and [tex]\lim_{x \to -\infty} 0[/tex]
The horizontal asymptote
Remove the denominator
f(x) = 30
Also, set the numerator to 0
f(x) = 0
Hence, the horizontal asymptotes are y = 30 and y = 0
The y-intercept
Set x = 0
[tex]f\left(0\right)=\frac{30}{1+2^{3-0}}[/tex]
Evaluate
[tex]f\left(0\right)=\frac{30}{9}[/tex]
Divide
[tex]f\left(0\right)=\frac{10}3[/tex]
Hence, the y-intercept is (0, 10/3)
How the numerator is related to (b)
The numerator of f(x) is 30.
In (b), we have
The horizontal asymptotes are y = 30 and y = 0
This means that the horizontal asymptotes and the numerator are the same
The graph
See attachment for the graph of f(x)
Read more about functions at:
https://brainly.com/question/12191454
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