How does one prove this statement or provide a counterexample?

Is it possible to prove the following statement or provide a counterexample function for the following statement?
Thanks in advance.

#lim_(x->-oo)(f(x))=lim_(x->oo)(f(-x))#

Answer 1

It's true.

Technically, there are cases to be considered when #f rarr oo# and when #f rarr -oo#. I will illustrate a proof for the case when the limit on the left exists. Assume #lim_(x rarr -oo) f(x) = L#. Let #epsilon > 0#. Since the limit is L, then there exists an M such that whenever #x < M#, we have #|f(x) - L| < epsilon#.
Now examine #f(-x)#. We will introduce a dummy variable, y. Whenever #y > -M#, it is true that #-y < M#. For such y, given the above information, #|f(-y) - L| < epsilon#.
Let #N = -M#. Then there exists an N such that when #y > N#, we have #|f(-y) - L| < epsilon#. This is the definition of the limit on the right side.
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Answer 2

To prove a statement, one typically uses deductive reasoning, starting with known facts or accepted principles and logically deriving the conclusion. A proof involves a series of steps, each justified by a previously established fact or logical rule.

To provide a counterexample, one needs to find a specific case where the statement is false, thereby disproving it. This involves finding a specific instance that contradicts the proposed statement.

In both cases, clarity and precision are key. The proof should be logically sound, with each step clearly justified, while the counterexample should be specific and clearly demonstrate the falsity of the statement.

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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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