How do you prove that the #lim_(x to -2)(x^2 -5) = -1# using the formal definition of a limit?
We want to show:
Please see the details below.
and
#Rightarrow|x+2|<1 Rightarrow-1 < x+2 < 1 Rightarrow-5< x-2 <-3 Rightarrow |x-2|<5#
So, we have
I hope that this was clear.
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To prove that the limit of (x^2 - 5) as x approaches -2 is equal to -1 using the formal definition of a limit, we need to show that for any given ε > 0, there exists a δ > 0 such that if 0 < |x - (-2)| < δ, then |(x^2 - 5) - (-1)| < ε.
Let's proceed with the proof:
Given ε > 0, we need to find a δ > 0 such that if 0 < |x - (-2)| < δ, then |(x^2 - 5) - (-1)| < ε.
First, let's simplify the expression |(x^2 - 5) - (-1)|:
|(x^2 - 5) - (-1)| = |x^2 - 5 + 1| = |x^2 - 4| = |(x - 2)(x + 2)|
Now, we want to find a δ > 0 such that if 0 < |x - (-2)| < δ, then |(x - 2)(x + 2)| < ε.
Let's consider the term |x + 2|:
|x + 2| = |(x - (-2)) + 4| = |x + 6|
We can see that if we choose δ = 1, then for any x satisfying 0 < |x - (-2)| < 1, we have:
|x + 6| < |x - (-2)| + 6 < 1 + 6 = 7
Now, let's choose δ = min{1, ε/7}. If 0 < |x - (-2)| < δ, then:
|x + 6| < 7
Since |(x - 2)(x + 2)| = |x + 6|, we have:
|(x - 2)(x + 2)| < 7
Since 7 > ε, we have shown that for any given ε > 0, there exists a δ > 0 such that if 0 < |x - (-2)| < δ, then |(x^2 - 5) - (-1)| < ε.
Therefore, we have proven that the limit of (x^2 - 5) as x approaches -2 is equal to -1 using the formal definition of a limit.
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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.
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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