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Left and right limits (One Sided Limits)

Look at the following graph of `f(x) = { (x, if x < 5), (2 , if x >= 5):}`


At x = 5, the graph is approaching different values from both sides. From left hand side of x = 5, the graph is approaching the value of 5 and from the right hand side the graph is approaching the value of 2.

Thus, the left hand limit of f(x) as x approaches 5 is said to be 5 and right hand limit as x approaches 5 is said to be 2. The left limit is denoted by a minus sign put on the right of the limit value as in `lim_(x -> 5^-) f(x) = 5` and the right limit is denoted by a plus sign put on the right of the limit value as `lim_(x -> 5^+) f(x) = 2`.
Thus there are two types of limits of a function. The left hand limit, and the right hand limit. These are collectively known as one sided limits.

When you are approaching a function towards a particular value of x, if you move in towards that particular value from its left hand side on the x axis, that is, from values just lesser than it, then you are finding the left hand limit. On the other hand if you approach the x value from the right hand side on the x axis, that is, from values greater than the particular value, you are finding the right hand limit.

The left and right limits of a function are important because many functions have different left and right limits for particular values of x, and, further, for many functions it is practically important to know just one of either  the left or right limits.

On combining the left and right limits of a function, you get the normal limit. This is the value that the function is approaching when you move in towards a particular x value from both the left and right hand sides on the x axis. When the left and right limits of a function are equal, it has a normal limit that is equal to them.

The above function has different left and right limits and so its normal limit does not exist.

Consider the following graph of `f(x) = -x^2 + 4x - 1`. For x = 2 the function exists and has a value of 3. The left hand limit as x approaches 2 is 3, because, as we move in towards x = 2 from the left hand side, that is, from values lesser than it on the x axis, the graph is approaching 3. Thus `lim_(x -> 2^-) f(x) = 3`

The right hand limit at x = 2 is 3, because, as we move in towards x = 2 from the right hand side, that is, from values greater than it on the x axis, the function's graph is approaching 3. Thus `lim_(x -> 2^+) f(x) = 3`

Since both the left and right hand limits of the function exist at x = 2 and are equal, so we can say that the normal limit of f(x) exists and is equal to 3. This is written as `lim_(x -> 2) f(x) = 3`

1 comment:

  1. It's a nice post about one sided limits. I really like it. It's really helpful. Thanks for sharing it.

    ReplyDelete