What is the limit of a piecewise function with different equations at x=2?

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In summary, the limit of g(x) is 1 as x approaches 2 from both the left and the right, as shown by the equations x^2-3 and cos(x-2).
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JessicaJ283782
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Homework Statement



Find the limit-> 2 g(x)= (x^2-3) if x < 2
3 if x=2
cos(x-2) if x>2

Homework Equations



So, I know you basically ignore the limit at 2, and you need to check it from the right and left. So, you want the x^2-3 and cos(x-2) equations

The Attempt at a Solution



I plugged two in for x, in both of these equations (2^2-3)=1 and cos(2-2)=cos(0)=1

So wouldn't the answer be 1?
 
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  • #2
JessicaJ283782 said:

Homework Statement



Find the limit-> 2 g(x)= (x^2-3) if x < 2
3 if x=2
cos(x-2) if x>2


Homework Equations



So, I know you basically ignore the limit at 2, and you need to check it from the right and left. So, you want the x^2-3 and cos(x-2) equations



The Attempt at a Solution



I plugged two in for x, in both of these equations (2^2-3)=1 and cos(2-2)=cos(0)=1

So wouldn't the answer be 1?

Yes, it would. Though you should be thinking 'the value of the functions approaches 1 as x->2' rather than saying 'I plugged in 2'.
 

1. What is the definition of a limit in calculus?

A limit in calculus is the value that a function approaches as the input values get closer and closer to a specific point, called the limit point. It is denoted by the notation "lim" and is used to understand the behavior of a function near a certain point.

2. How do you find the limit of a function?

To find the limit of a function, you must evaluate the function at the given limit point. If the function is continuous at that point, then the limit will be equal to the value of the function at that point. If the function is not continuous, then you must use algebraic manipulation or other techniques, such as L'Hopital's rule, to evaluate the limit.

3. What does it mean if the limit of a function does not exist?

If the limit of a function does not exist, it means that the function does not approach a single value as the input values get closer and closer to the limit point. This could be due to a jump or discontinuity in the function or the function approaching different values from opposite sides of the limit point.

4. Can a function have a limit at a point but not be defined at that point?

Yes, it is possible for a function to have a limit at a point where it is not defined. This occurs when the function has a removable discontinuity at that point. In this case, the limit would exist, but the function would need to be redefined at that point to make it continuous.

5. What is the difference between a one-sided limit and a two-sided limit?

A one-sided limit only considers the values of the function as it approaches the limit point from one direction (either the left or the right). A two-sided limit, on the other hand, considers the values of the function as it approaches the limit point from both the left and the right. Both types of limits can exist, but the two-sided limit must exist and be the same as both one-sided limits for the overall limit to exist.

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