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Check the continuity of f(x)= {{:((|x-4|...

Check the continuity of `f(x)= {{:((|x-4|)/(2(x-4)), if x ne 4),(0,if x = 4):}` at `x = 4`.

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To check the continuity of the function \( f(x) \) at \( x = 4 \), we need to verify the following condition: A function \( f(x) \) is continuous at \( x = a \) if: \[ \lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = f(a) \] In this case, \( a = 4 \). The function is defined as: \[ f(x) = \begin{cases} \frac{|x - 4|}{2(x - 4)} & \text{if } x \neq 4 \\ 0 & \text{if } x = 4 \end{cases} \] ### Step 1: Calculate the Left-Hand Limit We need to find \( \lim_{x \to 4^-} f(x) \): \[ \lim_{x \to 4^-} f(x) = \lim_{x \to 4^-} \frac{|x - 4|}{2(x - 4)} \] Since \( x \to 4^- \) means \( x < 4 \), we have \( |x - 4| = -(x - 4) = 4 - x \). Therefore, we can rewrite the limit as: \[ \lim_{x \to 4^-} \frac{4 - x}{2(x - 4)} = \lim_{x \to 4^-} \frac{-(x - 4)}{2(x - 4)} = \lim_{x \to 4^-} \frac{-1}{2} = -\frac{1}{2} \] ### Step 2: Calculate the Right-Hand Limit Next, we find \( \lim_{x \to 4^+} f(x) \): \[ \lim_{x \to 4^+} f(x) = \lim_{x \to 4^+} \frac{|x - 4|}{2(x - 4)} \] Since \( x \to 4^+ \) means \( x > 4 \), we have \( |x - 4| = x - 4 \). Thus, we can rewrite the limit as: \[ \lim_{x \to 4^+} \frac{x - 4}{2(x - 4)} = \lim_{x \to 4^+} \frac{1}{2} = \frac{1}{2} \] ### Step 3: Evaluate the Function at \( x = 4 \) Now we find the value of the function at \( x = 4 \): \[ f(4) = 0 \] ### Step 4: Compare the Limits and the Function Value Now we compare the left-hand limit, right-hand limit, and the function value: - Left-hand limit: \( \lim_{x \to 4^-} f(x) = -\frac{1}{2} \) - Right-hand limit: \( \lim_{x \to 4^+} f(x) = \frac{1}{2} \) - Function value: \( f(4) = 0 \) Since: \[ \lim_{x \to 4^-} f(x) \neq \lim_{x \to 4^+} f(x) \neq f(4) \] we conclude that the function \( f(x) \) is discontinuous at \( x = 4 \). ### Final Conclusion Thus, \( f(x) \) is discontinuous at \( x = 4 \). ---

To check the continuity of the function \( f(x) \) at \( x = 4 \), we need to verify the following condition: A function \( f(x) \) is continuous at \( x = a \) if: \[ \lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = f(a) \] In this case, \( a = 4 \). The function is defined as: ...
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