For the function f, of, xf(x) shown below, determine limit, start subscript, x, right arrow, 1, end subscript, f, of, x, . 
x→1
lim
​
 f(x).
f, of, x, equals, left curly bracket, begin cases, begin case 1, minus, 2, x, squared, minus, 3, for , x, is less than or equal to, 1, end case 1, begin case 2, minus, 4, plus, 4, x, for , x, is greater than, 1, end case 2, end cases, .
f(x)={ 
−2x 
2
 −3
−4+4x
​
  
 for x≤1 
1
1
​
 
 for x>1 
1
1
​
 
​


To find the limit of the function \( f(x) \) as \( x \to 1 \), we need to examine the behavior of the function from both the left-hand side (as \( x \) approaches 1 from values less than or equal to 1) and the right-hand side (as \( x \) approaches 1 from values greater than 1).

### The given piecewise function is:

\[
f(x) =
\begin{cases}
-2x^2 - 3 & \text{for } x \leq 1 \\
-4 + 4x & \text{for } x > 1
\end{cases}
\]

We will now compute the left-hand limit and right-hand limit of the function as \( x \to 1 \).

### 1. Left-hand limit (as \( x \to 1^- \)):

For \( x \leq 1 \), the function is \( f(x) = -2x^2 - 3 \).

Substitute \( x = 1 \) into this expression:

\[
f(1) = -2(1)^2 - 3 = -2 - 3 = -5
\]

Thus, the left-hand limit is:

\[
\lim_{x \to 1^-} f(x) = -5
\]

### 2. Right-hand limit (as \( x \to 1^+ \)):

For \( x > 1 \), the function is \( f(x) = -4 + 4x \).

Substitute \( x = 1 \) into this expression:

\[
f(1^+) = -4 + 4(1) = -4 + 4 = 0
\]

Thus, the right-hand limit is:

\[
\lim_{x \to 1^+} f(x) = 0
\]

### 3. Conclusion:

Since the left-hand limit \( (-5) \) and the right-hand limit \( (0) \) are not equal, the limit of \( f(x) \) as \( x \to 1 \) **does not exist**.

\[
\lim_{x \to 1} f(x) \quad \text{does not exist}.
\]

Would you like more details or have any questions on this problem?

Here are five related questions to expand your understanding:

1. How do you determine the continuity of a piecewise function?
2. What is the importance of matching left-hand and right-hand limits?
3. How would you calculate the limit of a trigonometric piecewise function?
4. Can a function be discontinuous at a point but have a defined limit?
5. How would you graph this piecewise function?

**Tip:** For piecewise functions, always check limits from both sides at the boundary points to determine if the function is continuous.

For the function f(x) shown below, determine lim x→1 f(x). f(x)={-2x^2 - 3 for x ≤ 1; -4 + 4x for x > 1}.

Determine the limit of the piecewise function f(x) as x approaches 1. This solution includes detailed calculations for the left-hand and right-hand limits, demonstrating that the limit does not exist. Suitable for Grades 10-12 students studying limits and continuity.

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