How do you use the epsilon delta definition to find the limit of #(2+4x)/3# as x approaches #1#?

Answer 1

First, a slightly technical point - you can't really find a limit using the #epsilon-delta# definition. What you can do is, given a number #L#, prove that it is the limit!

It seems obvious that

#lim_(x to 1) (2+4x)/3 = (2+4times 1)/3 =2#

Let us prove that 2, indeed, is the limit.

Recall that the #epsilon-delta# definition of the limit states that:
A number #L# is called the limit of a function #f : RR to RR# as #x to a# if #forall epsilon>0, exists delta >0# such that #0 < |x-a| < delta implies |f(x)-L| < epsilon#
Shorn of the Greek, this means that we can keep #f(x)# as close to #L# as we want, by keeping #x# sufficiently close to #a#.

For the current problem

#f(x)-L =(2+4x)/3-2=4/3(x-1)#
so #|f(x)-L|< epsilon iff 4/3|x-1| < epsilon#
So, for any given positive #epsilon#, choose
#0 < delta < 3/4epsilon#
Then, #|x-1| < delta implies |x-1| < 3/4epsilon implies |f(x)-L|=4/3 |x-1| < epsilon#
Hence such a #delta# exists for every positive #epsilon#, and thus 2 is the limit!
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Answer 2

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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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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