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Aug 9, 2024 · The limit doesn’t exist when the right and left sides of a function approach different values. If a function approaches either negative or positive infinity as it gets closer to a value, or if it oscillates between several values, the limit does not exist. Find where the limit doesn’t exist by graphing the function by hand or on a calculator.
Use the graph below to understand why $$\displaystyle\lim\limits_{x\to 3} f(x)$$ does not exist. In order for a limit to exist, the function has to approach a particular value. In the case shown above, the arrows on the function indicate that the the function becomes infinitely large. Since the function doesn't approach a particular value, the ...
Remember that limits represent the tendency of a function, so limits do not exist if we cannot determine the tendency of the function to a single point. Graphically, limits do not exist when: there is a jump discontinuity (Left-Hand Limit #ne# Right-Hand Limit) The limit does not exist at #x=1# in the graph below. there is a vertical asymptote
Oct 5, 2024 · If both one-sided limits exist and are equal, the two-sided limit exists. Cases When a Limit does not Exist. There are many possible cases when limit does not exist such as: Different Left-hand and Right-hand Limits; Unbounded Behavior; Oscillatory Behavior; Discontinuties; Let's discuss these in detail. Different Left-hand and Right-hand Limits
- Left Hand Limit Does Not Exist. In order for a limit to exist, both the left and right hand limits must exist, and they must have the same value. Here are some examples where the left hand limit does not exist.
- Right Hand Limit Does Not Exist. Just as a left hand limit can fail to exist, a right hand limit can also fail to exist. Here are some examples where the right hand limit does not exist.
- Left & Right Hand Limits Both Exist, But They Have Different Values. In some cases, both the left and right hand limits will exist for a function, but they will have different values.
- Function Is Not Defined Due To Domain Restriction. A limit can also fail to exist if a function is not defined due to a domain restriction. Example: Function Is Not Defined Due To Domain Restriction (Square Root)
Therefore, the limit doesn’t exist for y = x sin(1/x). However, you can make a good approximation with the Squeeze Theorem. Function settles on two different numbers. The function settles on more than one number as you move in towards your chosen x-value from the left and right.
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Nov 16, 2022 · Remember from the discussion after the first example that limits do not care what the function is actually doing at the point in question. Limits are only concerned with what is going on around the point. Since the only thing about the function that we actually changed was its behavior at \(x = 2\) this will not change the limit.
