Check the continuity of the following functions :
`f(x)=x^(2)" at "x=0`.

Answer

To check the continuity of the function \( f(x) = x^2 \) at \( x = 0 \), we need to verify three conditions:

1. The left-hand limit as \( x \) approaches 0.
2. The right-hand limit as \( x \) approaches 0.
3. The value of the function at \( x = 0 \).

If all three values are equal, then the function is continuous at that point.

### Step 1: Calculate the Left-Hand Limit
We start by calculating the left-hand limit as \( x \) approaches 0 from the left (denoted as \( 0^- \)):

\[ \lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} x^2 \]

To evaluate this limit, we can substitute \( x \) with \( 0 - h \) where \( h \) is a small positive number approaching 0:

\[ \lim_{h \to 0} (0 - h)^2 = \lim_{h \to 0} h^2 \]

As \( h \) approaches 0, \( h^2 \) also approaches 0:

\[ \lim_{h \to 0} h^2 = 0 \]

Thus, the left-hand limit is:

\[ \lim_{x \to 0^-} f(x) = 0 \]

### Step 2: Calculate the Right-Hand Limit
Next, we calculate the right-hand limit as \( x \) approaches 0 from the right (denoted as \( 0^+ \)):

\[ \lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} x^2 \]

Similarly, we substitute \( x \) with \( 0 + h \) where \( h \) is a small positive number approaching 0:

\[ \lim_{h \to 0} (0 + h)^2 = \lim_{h \to 0} h^2 \]

As \( h \) approaches 0, \( h^2 \) also approaches 0:

\[ \lim_{h \to 0} h^2 = 0 \]

Thus, the right-hand limit is:

\[ \lim_{x \to 0^+} f(x) = 0 \]

### Step 3: Calculate the Value of the Function at \( x = 0 \)
Now, we evaluate the function at \( x = 0 \):

\[ f(0) = 0^2 = 0 \]

### Conclusion
Now we have:

- Left-hand limit: \( \lim_{x \to 0^-} f(x) = 0 \)
- Right-hand limit: \( \lim_{x \to 0^+} f(x) = 0 \)
- Function value: \( f(0) = 0 \)

Since all three values are equal:

\[ \lim_{x \to 0^-} f(x) = \lim_{x \to 0^+} f(x) = f(0) = 0 \]

We conclude that the function \( f(x) = x^2 \) is continuous at \( x = 0 \).