How do you prove that the #lim_(x to -2)(x^2 -5) = -1# using the formal definition of a limit?

Answer 1

We want to show:

#forall epsilon>0#, #exists delta>0# s.t. #0<|x-(-2)|< delta Rightarrow |x^2-5-(-1)|< epsilon#

Please see the details below.

#forall epsilon>0#, #exists delta=min{epsilon/5,1}>0# s.t. #0<|x-(-2)|< delta#
#Rightarrow|x+2|< epsilon/5#

and

#Rightarrow|x+2|<1 Rightarrow-1 < x+2 < 1 Rightarrow-5< x-2 <-3 Rightarrow |x-2|<5#

So, we have

#|x^2-5-(-1)|=|x^2-4|=|x+2||x-2|< epsilon/5 cdot 5=epsilon#
Hence, #lim_(x to -2)(x^2-5)=-1#

I hope that this was clear.

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Answer 2

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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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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