If `f(x)={(x^2+1", "x ne 1),(" "3 ", "x=1):}` , then check whether the function f(x) is continuous or discontinuous at x=1

Answer

To determine whether the function \( f(x) \) is continuous or discontinuous at \( x = 1 \), we will follow these steps:

### Step 1: Identify the function
The function is defined as:
\[ f(x) = \begin{cases} x^2 + 1 & \text{if } x \neq 1 \\ 3 & \text{if } x = 1 \end{cases} \]

### Step 2: Calculate the limit as \( x \) approaches 1
We need to find:
\[ \lim_{x \to 1} f(x) \]
Since we are approaching \( x = 1 \) but not equal to 1, we will use the first part of the function:
\[ \lim_{x \to 1} f(x) = \lim_{x \to 1} (x^2 + 1) \]
Now, substituting \( x = 1 \):
\[ = 1^2 + 1 = 1 + 1 = 2 \]

### Step 3: Find the value of the function at \( x = 1 \)
Now, we find \( f(1) \):
\[ f(1) = 3 \]

### Step 4: Compare the limit and the function value
We have:
\[ \lim_{x \to 1} f(x) = 2 \quad \text{and} \quad f(1) = 3 \]
Since these two values are not equal:
\[ \lim_{x \to 1} f(x) \neq f(1) \]

### Step 5: Conclusion
Since the limit of \( f(x) \) as \( x \) approaches 1 is not equal to the value of the function at \( x = 1 \), we conclude that:
\[ f(x) \text{ is discontinuous at } x = 1. \]

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