Find `lim_(x to 1) f(x)`, where `f(x) = {{:(x + 1, x != 1),(0, x = 1):}}`

Answer

To find the limit \( \lim_{x \to 1} f(x) \), where

\[ f(x) = \begin{cases} x + 1 & \text{if } x \neq 1 \\ 0 & \text{if } x = 1 \end{cases} \]

we will evaluate the limit by considering the right-hand limit and the left-hand limit as \( x \) approaches 1.

### Step 1: Find the Right-Hand Limit
We start by calculating the right-hand limit, denoted as \( \lim_{x \to 1^+} f(x) \). Since we are approaching 1 from the right, \( x \) will be slightly greater than 1. In this case, we use the definition of \( f(x) \) for \( x \neq 1 \):

\[ \lim_{x \to 1^+} f(x) = \lim_{x \to 1^+} (x + 1) \]

Substituting \( x = 1 \):

\[ \lim_{x \to 1^+} (x + 1) = 1 + 1 = 2 \]

### Step 2: Find the Left-Hand Limit
Next, we calculate the left-hand limit, denoted as \( \lim_{x \to 1^-} f(x) \). Here, \( x \) will be slightly less than 1. Again, we use the definition of \( f(x) \) for \( x \neq 1 \):

\[ \lim_{x \to 1^-} f(x) = \lim_{x \to 1^-} (x + 1) \]

Substituting \( x = 1 \):

\[ \lim_{x \to 1^-} (x + 1) = 1 + 1 = 2 \]

### Step 3: Conclude the Limit
Since both the right-hand limit and the left-hand limit are equal:

\[ \lim_{x \to 1^+} f(x) = 2 \quad \text{and} \quad \lim_{x \to 1^-} f(x) = 2 \]

We can conclude that:

\[ \lim_{x \to 1} f(x) = 2 \]

### Final Answer
Thus, the limit is:

\[ \lim_{x \to 1} f(x) = 2 \]

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