Questions: 1. Let f(x)=(x-2)/x-2, x ≠ 2 (a) Sketch the graph of f. (b) Explain why lim x → 2 does not exist. Your explanation should be in terms of appropriate limits. (c) Is f continuous at 2? Explain. 2. Explain why lim t → 4 sqrt(4-t) does not exist.

1. Let f(x)=(x-2)/x-2, x ≠ 2
(a) Sketch the graph of f.
(b) Explain why lim x → 2 does not exist. Your explanation should be in terms of appropriate limits.
(c) Is f continuous at 2? Explain.
2. Explain why lim t → 4 sqrt(4-t) does not exist.

Transcript text: 1. Let $f(x)=\frac{x-2}{|x-2|}, x \neq 2$ (a) Sketch the graph of $f$. (b) Explain why $\lim _{x \rightarrow 2}$ does not exist. Your explanation should be in terms of appropriate limits. (c) Is $f$ continuous at 2? Explain. 2. Explain why $\lim _{t \rightarrow 4} \sqrt{4-t}$ does not exist.

Solution

Solution Steps

Step 1: Define the function $ f(x) $

The function is given by: \[ f(x) = \frac{x-2}{|x-2|}, \quad x \neq 2 \]

Step 2: Sketch the graph of $ f(x) $

The function $ f(x) $ can be simplified based on the value of $ x $:

For $ x > 2 $, $ f(x) = 1 $
For $ x < 2 $, $ f(x) = -1 $

Step 3: Explain why $ \lim_{x \to 2} f(x) $ does not exist

To determine if the limit exists, we need to check the left-hand limit and the right-hand limit:

Left-hand limit: $ \lim_{x \to 2^-} f(x) = -1 $
Right-hand limit: $ \lim_{x \to 2^+} f(x) = 1 $

Since the left-hand limit and the right-hand limit are not equal, the limit $ \lim_{x \to 2} f(x) $ does not exist.

Final Answer

(a) The graph of $ f(x) $ is a piecewise function with $ f(x) = 1 $ for $ x > 2 $ and $ f(x) = -1 $ for $ x < 2 $.

(b) The limit $ \lim_{x \to 2} f(x) $ does not exist because the left-hand limit and the right-hand limit are not equal: \[ \lim_{x \to 2^-} f(x) = -1 \] \[ \lim_{x \to 2^+} f(x) = 1 \]

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