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Sagot :
To determine whether the given function is continuous, we need to check the continuity at the point where the function changes its definition, which is at [tex]\( x = -2 \)[/tex]. The function is defined as:
[tex]\[ f(x) = \begin{cases} x - 3 & \text{if } x \leq -2 \\ 4x + 5 & \text{if } x > -2 \end{cases} \][/tex]
### Step-by-Step Solution:
1. Determine [tex]\( \lim_{x \to -2^-} f(x) \)[/tex]:
For [tex]\( x \leq -2 \)[/tex], the function is [tex]\( f(x) = x - 3 \)[/tex]. We need to find the limit of [tex]\( f(x) \)[/tex] as [tex]\( x \)[/tex] approaches [tex]\(-2\)[/tex] from the left.
[tex]\[ \lim_{x \to -2^-} f(x) = \lim_{x \to -2^-} (x - 3) = -2 - 3 = -5 \][/tex]
2. Determine [tex]\( \lim_{x \to -2^+} f(x) \)[/tex]:
For [tex]\( x > -2 \)[/tex], the function is [tex]\( f(x) = 4x + 5 \)[/tex]. We need to find the limit of [tex]\( f(x) \)[/tex] as [tex]\( x \)[/tex] approaches [tex]\(-2\)[/tex] from the right.
[tex]\[ \lim_{x \to -2^+} f(x) = \lim_{x \to -2^+} (4x + 5) = 4(-2) + 5 = -8 + 5 = -3 \][/tex]
3. Evaluate [tex]\( f(-2) \)[/tex]:
For [tex]\( x = -2 \)[/tex], since [tex]\( x \leq -2 \)[/tex], we use the definition [tex]\( f(x) = x - 3 \)[/tex]:
[tex]\[ f(-2) = -2 - 3 = -5 \][/tex]
4. Check Continuity at [tex]\( x = -2 \)[/tex]:
For the function [tex]\( f(x) \)[/tex] to be continuous at [tex]\( x = -2 \)[/tex], the left-hand limit, right-hand limit, and the function value at [tex]\( x = -2 \)[/tex] must all be equal.
[tex]\[ \lim_{x \to -2^-} f(x) = -5 \][/tex]
[tex]\[ \lim_{x \to -2^+} f(x) = -3 \][/tex]
[tex]\[ f(-2) = -5 \][/tex]
Since [tex]\( \lim_{x \to -2^-} f(x) \neq \lim_{x \to -2^+} f(x) \)[/tex], the limits are not equal. Therefore, the function [tex]\( f(x) \)[/tex] is not continuous at [tex]\( x = -2 \)[/tex].
### Graphing the Function:
To help visualize the function, consider the graph where the function changes its definition at [tex]\( x = -2 \)[/tex]:
- For [tex]\( x \leq -2 \)[/tex], the function follows the line [tex]\( y = x - 3 \)[/tex].
- For [tex]\( x > -2 \)[/tex], the function follows the line [tex]\( y = 4x + 5 \)[/tex].
At [tex]\( x = -2 \)[/tex], we have a break in the graph since the value of the function jumps from [tex]\( -5 \)[/tex] to [tex]\( -3 \)[/tex].
### Conclusion:
Based on the above analysis, the function [tex]\( f \)[/tex] is not continuous at [tex]\( x = -2 \)[/tex].
Therefore, the function is not continuous.
The answer is No.
[tex]\[ f(x) = \begin{cases} x - 3 & \text{if } x \leq -2 \\ 4x + 5 & \text{if } x > -2 \end{cases} \][/tex]
### Step-by-Step Solution:
1. Determine [tex]\( \lim_{x \to -2^-} f(x) \)[/tex]:
For [tex]\( x \leq -2 \)[/tex], the function is [tex]\( f(x) = x - 3 \)[/tex]. We need to find the limit of [tex]\( f(x) \)[/tex] as [tex]\( x \)[/tex] approaches [tex]\(-2\)[/tex] from the left.
[tex]\[ \lim_{x \to -2^-} f(x) = \lim_{x \to -2^-} (x - 3) = -2 - 3 = -5 \][/tex]
2. Determine [tex]\( \lim_{x \to -2^+} f(x) \)[/tex]:
For [tex]\( x > -2 \)[/tex], the function is [tex]\( f(x) = 4x + 5 \)[/tex]. We need to find the limit of [tex]\( f(x) \)[/tex] as [tex]\( x \)[/tex] approaches [tex]\(-2\)[/tex] from the right.
[tex]\[ \lim_{x \to -2^+} f(x) = \lim_{x \to -2^+} (4x + 5) = 4(-2) + 5 = -8 + 5 = -3 \][/tex]
3. Evaluate [tex]\( f(-2) \)[/tex]:
For [tex]\( x = -2 \)[/tex], since [tex]\( x \leq -2 \)[/tex], we use the definition [tex]\( f(x) = x - 3 \)[/tex]:
[tex]\[ f(-2) = -2 - 3 = -5 \][/tex]
4. Check Continuity at [tex]\( x = -2 \)[/tex]:
For the function [tex]\( f(x) \)[/tex] to be continuous at [tex]\( x = -2 \)[/tex], the left-hand limit, right-hand limit, and the function value at [tex]\( x = -2 \)[/tex] must all be equal.
[tex]\[ \lim_{x \to -2^-} f(x) = -5 \][/tex]
[tex]\[ \lim_{x \to -2^+} f(x) = -3 \][/tex]
[tex]\[ f(-2) = -5 \][/tex]
Since [tex]\( \lim_{x \to -2^-} f(x) \neq \lim_{x \to -2^+} f(x) \)[/tex], the limits are not equal. Therefore, the function [tex]\( f(x) \)[/tex] is not continuous at [tex]\( x = -2 \)[/tex].
### Graphing the Function:
To help visualize the function, consider the graph where the function changes its definition at [tex]\( x = -2 \)[/tex]:
- For [tex]\( x \leq -2 \)[/tex], the function follows the line [tex]\( y = x - 3 \)[/tex].
- For [tex]\( x > -2 \)[/tex], the function follows the line [tex]\( y = 4x + 5 \)[/tex].
At [tex]\( x = -2 \)[/tex], we have a break in the graph since the value of the function jumps from [tex]\( -5 \)[/tex] to [tex]\( -3 \)[/tex].
### Conclusion:
Based on the above analysis, the function [tex]\( f \)[/tex] is not continuous at [tex]\( x = -2 \)[/tex].
Therefore, the function is not continuous.
The answer is No.
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