Question

What is a left-hand limit?

Answer 1

A left-hand limit means the limit of a function as it approaches from the left-hand side.

On the other hand, A right-hand limit means the limit of a function as it approaches from the right-hand side.

When getting the limit of a function as it approaches a number, the idea is to check the behavior of the function as it approaches the number. We substitute values as close as possible to the number being approached.

The closest number is the number being approached itself. Hence, one usually just substitutes the number being approached to get the limit.

However, we cannot do this if the resulting value is undefined.
But we can still check its behavior as it approaches from one side.

One good example is `lim_(x->0) 1/x`.

When we substitute `x = 0` into the function, the resulting value is undefined.

Let's check its limit as it approaches from the left-hand side

`f(x) = 1/x`

`f(-1) = 1/-1 = -1`
`f(-1/2) = 1/(-1/2) = -2`
`f(-1/10) = 1/(-1/10) = -10`
`f(-1/1000) = 1/(-1/1000) = -1000`
`f(-1/1000000) = 1/(-1/1000000) = -1000000`

Notice that as we get closer and closer to `x = 0` from the left-hand side, the resulting value we gets larger and larger (though negative). We can conclude that the limit as `x -> 0` from the left-hand side is `-oo`

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Now let's check the limit from the right-hand side

`f(x) = 1/x`

`f(1) = 1/1 = 1`
`f(1/2) = 1/(1/2) = 2`
`f(1/10) = 1/(1/10) = 10`
`f(1/1000) = 1/(1/1000) = 1000`
`f(1/1000000) = 1/(1/1000000) = 1000000`

The limit as `x -> 0` from the right-hand side is `oo`

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When the left-hand side limit of a function is different from the right-hand side limit, we can conclude that the function is discontinuous at the number being approached.