How do you find discontinuity algebraically?
There is no universal method that works for all possible functions.
The problems beginning calculus students are presented usually involve either:
Because the left and right limits are equa, we have:
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To find discontinuity algebraically, you need to analyze the function's behavior at the point in question. Look for any of the following conditions:

Removable Discontinuity: Check if the function is undefined at a specific point but can be made continuous by redefining the function at that point.

Jump Discontinuity: Determine if the function has different finite limits from the left and right sides of the point.

Infinite Discontinuity: Verify if the function approaches positive or negative infinity as it approaches the point.

Oscillating Discontinuity: Identify if the function oscillates infinitely as it approaches the point.
To summarize, analyze the function's behavior at the point to determine if it exhibits any of the mentioned types of discontinuity.
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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