Find the limits as x approaches 1 from the left and right side for f(x) = (x^2 - 3, 0≤*x*<1), (-1, x=1), (3x - 5, 1<*x*)

We are tasked with finding the left-hand limit (\(\lim_{x \to 1^-} f(x)\)) and the right-hand limit (\(\lim_{x \to 1^+} f(x)\)) for the given piecewise function:  

\[
f(x) = 
\begin{cases} 
x^2 - 3, & 0 \leq x < 1 \\
-1, & x = 1 \\
3x - 5, & x > 1 
\end{cases}
\]

### Step 1: Evaluate \(\lim_{x \to 1^-} f(x)\)
As \(x \to 1^-\), the function corresponds to the first piece \(f(x) = x^2 - 3\).  
Substituting \(x = 1\) into \(x^2 - 3\), we get:
\[
\lim_{x \to 1^-} f(x) = 1^2 - 3 = -2.
\]

### Step 2: Evaluate \(\lim_{x \to 1^+} f(x)\)
As \(x \to 1^+\), the function corresponds to the third piece \(f(x) = 3x - 5\).  
Substituting \(x = 1\) into \(3x - 5\), we get:
\[
\lim_{x \to 1^+} f(x) = 3(1) - 5 = -2.
\]

### Step 3: Compare the left-hand and right-hand limits
Since both the left-hand and right-hand limits are equal (\(-2\)), the two-sided limit exists and is given by:
\[
\lim_{x \to 1} f(x) = -2.
\]

### Step 4: Note about the value of \(f(x)\) at \(x = 1\)
The function is explicitly defined as \(f(1) = -1\), which is different from the limit as \(x \to 1\). Therefore, \(f(x)\) has a discontinuity at \(x = 1\).

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### Final Answer:
\[
\lim_{x \to 1^-} f(x) = -2, \quad \lim_{x \to 1^+} f(x) = -2.
\]

Would you like a detailed graph or further clarification? 😊

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### Related Questions:
1. What are the different types of discontinuities in piecewise functions?
2. How do you determine if a function is continuous at a point?
3. Can a two-sided limit exist if the left and right limits differ?
4. How do you graph piecewise functions?
5. What is the difference between a limit and the value of a function at a point?

### Tip:
Always check each piece of a piecewise function carefully when evaluating limits, as the behavior of the function may change at the boundary points.

Find the limits as x approaches 1 from the left and right side for f(x) = (x^2 - 3, 0≤x<1), (-1, x=1), (3x - 5, 1<x).

Learn how to evaluate the left-hand and right-hand limits of a piecewise function at x = 1, using the function f(x) = (x^2 - 3, 0≤x<1), (-1, x=1), (3x - 5, 1<x). This problem explains the calculation of limits, their comparison, and discontinuity analysis, suitable for Grades 10-12.

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