Answer :
The value of the value of limit [ff(x))] will be: [tex]\lim _{x \rightarrow 2} f(f(x))=-1[/tex]
A limit in mathematics is the value that a function, output, or sequence approaches when an input, index, or other quantity approaches a certain value. Calculus and mathematical analysis are not possible without limits, which are also required to determine continuity, derivatives, and integrals.
By evaluating the function at values close to x=0, we may determine the size of a limit, if one exists. We are unable to immediately determine a function value for x=0 since the outcome would have an unknown denominator.
We have been given
[tex]$$f(x)=\left\{\begin{array}{cc}-x^2+3 x+3 & \text { for } x < 2 \\6 & \text { for } x=2 \\8-\frac{3}{2} x & \text { for } x > 2\end{array}\right.$$[/tex]
We need to find the value of [tex]$\lim _{\lambda \rightarrow 2} f(f(x))$[/tex]
We know [tex]\lim _{x \rightarrow 2} f(x)=f(2)=6[/tex]
Now [tex]$\lim _{x \rightarrow 2} f[f(x)]=\lim _{x \rightarrow 2} f(6)$[/tex]=f(6)
We can find from given range as follows:
[tex]f(6)=8-\frac{3}{2} x\\=8-\frac{3}{2} \times 6\\=8-9=-1[/tex]
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Final answer:
The question is asking for the limit of a piecewise function as x approaches 2. After showing that the limit from the right exists and equals 6, and that this is the same as the value of the function at x=2, we can conclude that the limit as x approaches 2 is 6.
Explanation:
This question is about evaluating limits of a piecewise function. If we denote the limit as we approach a value from the right as lim+ and the limit from the left as lim-, then the overall limit will exist only if both of these limits exist and are equal.
Looking at the piecewise function, for x less than 2, the function is given by f(x) = --- + 3x + 3 but this part of the function is irrelevant to the problem at hand as we are interested in the limit around x = 2.
At x = 2, the function becomes simply 6, but this doesn't necessarily determine the limit.
For x greater than 2, the function is given by f(x) = 8 - x. The limit as we approach 2 from the right (or lim+) is thus 8 - 2 which equals 6.
Since f(2) = 6, and the limit as we approach 2 from the right is also 6, we conclude that lim f(x) as x approaches 2 is 6.
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