Find the limit $L$ (if it exists). (If the limit is infinite, enter '$\infty$' or '-$\infty$', as appropriate. If the limit does not otherwise exist, enter DNE.)
[tex]
\begin{array}{l}
\lim _{x \rightarrow 6} f(x) \text { where } f(x)=\left{\begin{array}{ll}
(x-6)^2, & x \leq 6 \\
6-x, & x\ \textgreater \ 6
\end{array}\right. \\
L=\square
\end{array}
[/tex]
If it does not exist, explain why.
A. The limit does not exist because the function does not approach $f(6)$ as $x$ approaches 6.
B. The limit does not exist because the function value is undefined at $x=6$.
C. The limit does not exist because the function is not continuous at any $x$ value.
D. The limit does not exist because the function approaches different values from the left and right side of 6.
E. The limit exists.
Answer (2)
Calculate the left-hand limit as x approaches 6: lim x → 6 − ( x − 6 ) 2 = 0 .
Calculate the right-hand limit as x approaches 6: lim x → 6 + ( 6 − x ) = 0 .
Since the left-hand limit and the right-hand limit are equal, the limit exists.
The limit of the function as x approaches 6 is 0 .
Explanation
Problem Analysis We are given a piecewise function f ( x ) and asked to find the limit as x approaches 6. Since the function is defined differently for x l e q 6 and 6"> x > 6 , we need to evaluate the left-hand limit and the right-hand limit separately.
Left-Hand Limit The left-hand limit is the limit as x approaches 6 from the left, i.e., x < 6 . In this case, f ( x ) = ( x − 6 ) 2 . So, we have l i m x t o 6 − f ( x ) = l i m x t o 6 − ( x − 6 ) 2
Calculating Left-Hand Limit Substituting x = 6 into the expression ( x − 6 ) 2 , we get ( 6 − 6 ) 2 = 0 2 = 0 . Therefore, the left-hand limit is 0.
Right-Hand Limit The right-hand limit is the limit as x approaches 6 from the right, i.e., 6"> x > 6 . In this case, f ( x ) = 6 − x . So, we have l i m x t o 6 + f ( x ) = l i m x t o 6 + ( 6 − x )
Calculating Right-Hand Limit Substituting x = 6 into the expression ( 6 − x ) , we get 6 − 6 = 0 . Therefore, the right-hand limit is 0.
Comparing Limits and Conclusion Since the left-hand limit and the right-hand limit are both equal to 0, the limit exists and is equal to 0. l i m x t o 6 f ( x ) = 0
Final Answer The limit of the function f ( x ) as x approaches 6 is 0.
Examples
Imagine you're designing a ramp where the height is defined by a piecewise function. As you approach a certain point on the ramp (in this case, x = 6 ), you need to ensure the height smoothly transitions to avoid any sudden jumps. Calculating the limit at that point, as we did here, confirms whether the two parts of the ramp connect seamlessly, ensuring a safe and continuous path.
The limit of the function f ( x ) as x approaches 6 exists and is equal to 0. Both the left-hand and right-hand limits approach 0. Therefore, L = 0 .
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