Find `lim_(X to 0) f(x)` where `f(x) = {{:(x, x!=0),(5,x=0):}}`

Answer

To find the limit of the function \( f(x) \) as \( x \) approaches 0, we can follow these steps:

### Step 1: Define the Function
The function \( f(x) \) is defined as:
\[ f(x) = \begin{cases} x & \text{if } x \neq 0 \\ 5 & \text{if } x = 0 \end{cases} \]

### Step 2: Find the Right-Hand Limit
To find the right-hand limit as \( x \) approaches 0, we calculate:
\[ \lim_{x \to 0^+} f(x) \]
Since \( x \) is approaching 0 from the right, \( x \) is not equal to 0, so we use the first case of the function:
\[ \lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} x = 0 \]

### Step 3: Find the Left-Hand Limit
Next, we find the left-hand limit as \( x \) approaches 0:
\[ \lim_{x \to 0^-} f(x) \]
Similarly, since \( x \) is approaching 0 from the left, \( x \) is still not equal to 0, so we again use the first case of the function:
\[ \lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} x = 0 \]

### Step 4: Compare the Limits
Now we compare the right-hand limit and the left-hand limit:
\[ \lim_{x \to 0^+} f(x) = 0 \quad \text{and} \quad \lim_{x \to 0^-} f(x) = 0 \]
Since both limits are equal, we can conclude that:
\[ \lim_{x \to 0} f(x) = 0 \]

### Final Answer
Thus, the limit is:
\[ \lim_{x \to 0} f(x) = 0 \]

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