How do you use the epsilon delta definition to find the limit of #(2+4x)/3# as x approaches #1#?
First, a slightly technical point - you can't really find a limit using the
It seems obvious that
Let us prove that 2, indeed, is the limit.
For the current problem
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To use the epsilon-delta definition to find the limit of (2+4x)/3 as x approaches 1, we need to show that for any given epsilon (ε) greater than 0, there exists a delta (δ) greater than 0 such that if 0 < |x - 1| < δ, then |(2+4x)/3 - L| < ε, where L is the limit we are trying to find.
Let's proceed with the calculation:
Given the expression (2+4x)/3, we can simplify it to (4x+2)/3.
Now, let's find the value of L by substituting x = 1 into the expression: L = (4(1)+2)/3 = 6/3 = 2.
Next, we need to find a suitable delta (δ) for a given epsilon (ε).
Let's start by manipulating the inequality |(2+4x)/3 - 2| < ε:
|(2+4x)/3 - 2| < ε |(2+4x)/3 - 6/3| < ε |(2+4x-6)/3| < ε |4x-4|/3 < ε 4|x-1|/3 < ε
Now, we can set 4|x-1|/3 < ε and solve for δ:
4|x-1|/3 < ε |x-1| < 3ε/4
From here, we can see that if we choose δ = 3ε/4, then the inequality holds.
Therefore, for any given ε > 0, if 0 < |x - 1| < δ = 3ε/4, then |(2+4x)/3 - 2| < ε.
Hence, the limit of (2+4x)/3 as x approaches 1 is 2.
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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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