How to determine if a limit is continuous or not
To determine if a limit is continuous or not, you need to check three conditions:
1. The limit of the function as x approaches the point where the limit is being evaluated (let's call it "a") must exist. This means that the left-hand and right-hand limits must be the same.
2. The function must be defined at the point "a".
3. The limit of the function as x approaches "a" must be equal to the function value at "a".
If all three conditions are true, then the limit is continuous. If any of the conditions are false, then the limit is not continuous.
To determine if a limit is continuous or not, you need to consider the following steps:
1. Evaluate the limit: Calculate the value of the limit by substituting the given value(s) into the function. This is the first step and will determine whether or not the limit exists.
2. Check if the function is defined at the point: To be continuous, a function must be defined at the value where the limit is being calculated. If the function is not defined at that point, then it is definitely not continuous.
3. Compare the limit and the function value: If the limit exists and the function is defined at the given point, compare the value of the limit and the function value. If they are equal, then the function is continuous at that point.
4. Check for removable discontinuity: If the limit exists, but the function is not defined or does not equal the limit value, it might have a removable discontinuity. This occurs when there's a hole in the graph that can be filled in by adjusting a single function value.
5. Identify jump or infinite discontinuity: If the limit does not exist or is not equal to the function value, and the function is defined at that point, it may have a jump discontinuity. This happens when the left and right limits are finite, but not equal.
6. Determine if there are asymptotes: If the limit does not exist because it approaches infinity or negative infinity, or if the function value is infinite, there might be horizontal, vertical, or slant asymptotes.
By following these steps, you can determine whether a limit is continuous or not.