7. When Limits Do Not Exist (DNE)

A limit does not exist when: • The function approaches different values from the left and right. • The function oscillates infinitely near the point. • The function approaches infinity or negative infinity.

In calculus, we say that a limit does not exist (DNE) under certain conditions. The first condition is when the values of the function approach different numbers as you come from the left side and the right side of a point. This means the expected limit cannot be determined since both sides provide conflicting values. The second condition is when a function continuously oscillates near a certain point, making it impossible to assign a single limit to that point. Lastly, if a function grows indefinitely large (positive infinity) or indefinitely small (negative infinity) as it approaches a certain point, we say the limit does not exist since the values are not finite.

Imagine you're at a busy intersection trying to get through but cars keep coming from both directions. If you're trying to cross, but traffic lights are broken, and the cars on the left and right are moving in different ways, you can't just step out safely into the street; similar to how a limit can't exist when the function behaves differently from either side.

• The function approaches different values from the left and right.

This chunk explains that a limit does not exist when the function output differs when approaching from the left side compared to the right. For example, if approaching from the left you get a value of 3 and from the right a value of 5, the limit cannot be definitively set since both sides disagree. Formally, we express this in terms of the left-hand limit and right-hand limit; if these values do not match, the limit is considered to not exist at that point.

Consider a game of tug-of-war where one team pulls hard from the left while another team pulls from the right. If both teams are pulling with different levels of force, the rope (representing the limit) doesn’t settle at a specific point, just like the limit of the function can't settle at a specific value if the sides differ.

• The function oscillates infinitely near the point.

In this case, a limit does not exist when the function oscillates infinitely as it approaches a point. This means the outputs of the function do not settle toward a particular value as you get closer to the point; instead, they bounce back and forth without converging. This erratic behavior makes it impossible to assign a single limit value at that point, like trying to define a precise location when a person is rapidly moving back and forth.

Think of trying to catch a butterfly in a garden. The butterfly flits around erratically, moving quickly in different directions rather than landing. If you're trying to estimate where it's going to be next, it’s incredibly hard, just like how we can’t find a consistent limit where the function doesn't settle down but keeps oscillating instead.

• The function approaches infinity or negative infinity.

This part refers to situations where as the variable approaches a certain point, the function itself has values that continue to grow larger and larger, or drop indefinitely towards negative values instead of approaching a finite number. When this happens, we say that the limit does not exist because it does not approach a single numerical value but rather diverges to infinity or negative infinity.

Imagine a hot air balloon that is rising high into the sky without limits. The higher it goes, the further off the charts it climbs. Instead of settling at a number (like a height that you could give a measurement), it keeps going up endlessly or could metaphorically represent going down as well. Just like in math, if the value keeps going up or down without stopping, we can't define it at a specific point.

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