# Why is it impossible to have #lim_(x->0) f(x)# and #lim_(f(x)->0)f(x)# simultaneously exist for any of these graphs?

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#A)# #f(x) = 1/x^2#

#B)# #f(x) = -1/x^2#

#C)# #f(x) = 1/x#

#D)# #f(x) = -1/x#

All four of these graphs have the

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For the function

(Imagine trying to run to two different places at once; can't do it.)

Both kinds of asymptotes are on the graph, to be sure, but you can only approach one of those kinds of asymptotes at a time.

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It is impossible to have both lim_(x->0) f(x) and lim_(f(x)->0) f(x) simultaneously exist for any of these graphs because the limit as x approaches 0 and the limit as f(x) approaches 0 are two different concepts. The limit as x approaches 0 refers to the behavior of the function as x gets arbitrarily close to 0, while the limit as f(x) approaches 0 refers to the behavior of the function as the output values approach 0. These two limits are not necessarily related and can have different outcomes.

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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.

- How do you evaluate the limit #-w^2+1# as w approaches #-2#?
- What is the limit of #[arctan(e^b) - pi/4]# as b approaches infinity?
- evaluate # lim_(x rarr 0) (sin3x-sinx)/sinx #?
- How do you evaluate the limit #(2-x)/(x^2-4)# as x approaches #2#?
- How do you find the limit of #(3 sin (x)(1 - cos(x))) / (x^2) # as x approaches 0?

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