How do you find one sided limits and limits of piecewise functions?

Answer 1

With piecewise defined functions, you only have to worry about selecting the correct formula, as there are several possible combinations.

Let's examine an illustration.

#f(x)={(x+2 if x<-2),(x^2 if -2 le x <1),(1/x if 1 le x):}#
(1) #lim_{x to -2^-}f(x)=lim_{x to -2^-}(x+2)=-2+2=0#
(2) #lim_{x to -2^+}f(x)=lim_{x to -2^+}x^2=(-2)^2=4#
(3) #lim_{x to -2}f(x)# does not exist since #(1) ne (2)#
(4) #lim_{x to 1^-}f(x)=lim_{x to 1^-}x^2=(1)^2=1#
(5) #lim_{x to 1^+}f(x)=lim_{x to 1^+}1/x=1/1=1#
(6) #lim_{x to 1}f(x)=1# since (#(4)=(5)#)

As you can see above, a two-sided limit occurs only when the left- and right-hand limits are equal. You just need to select the appropriate formula based on which direction the limit is approaching from.

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Answer 2

To find one-sided limits of a function, evaluate the function as the input approaches the given value from the left or right side. For example, to find the left-hand limit, substitute values slightly less than the given value into the function and observe the resulting outputs. Similarly, to find the right-hand limit, substitute values slightly greater than the given value into the function and observe the outputs.

To find limits of piecewise functions, evaluate each piece of the function separately as the input approaches the given value. Determine the left-hand and right-hand limits for each piece, and if they are equal, that value is the limit. If the left-hand and right-hand limits differ, the limit does not exist.

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Answer from HIX Tutor

When evaluating a one-sided 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 one-sided 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 one-sided 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 one-sided 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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