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![Determine whether the function
f(x) is continuous at
x=-4.
f(x)={[1+x^(2)",",x >= 4],[7+2x",",x < 4]:}
f(x) is discontinuous at
x=-4
f(x) is continuous at
x=-4](https://externalquestionsimg-a4fuc2b6e0gtdqge.z01.azurefd.net/external-question-images/images/images/dm/Calculus/Continuity/Continuity/Q-35.png)
Determine whether the function is continuous at . is discontinuous at is continuous at
Full solution
Q. Determine whether the function is continuous at . is discontinuous at is continuous at
- Definition of Continuity: Understand the definition of continuity at a point.A function is continuous at a point if the following three conditions are met:. is defined.. The limit of as approaches exists.. The limit of as approaches is equal to .
- Check Defined: Check if is defined.Since is less than , we use the piece of the function defined for x < 4, which is .. is defined and equals .
- Left Limit at : Check the limit of as approaches from the left.Since we are approaching from the left, we are still in the region where x < 4, so we use the same piece of the function, ..The limit from the left exists and equals .
- Right Limit at : Check the limit of as approaches from the right.Since we are approaching from the right, we are still in the region where x < 4, so we use the same piece of the function, ..The limit from the right exists and equals .
- Compare Left and Right Limits: Determine if the left-hand limit and right-hand limit are equal.From Steps and , we have:Since both limits are equal, the limit of as approaches exists and equals .
- Comparison of and Limit: Compare the value of with the limit of as approaches . From Step , we have . From Step , we have . Since is equal to the limit of as approaches , the function is continuous at .
