I don't know limits fairly well. The problem is prove the limit using definition 6,
lim x -> -3, 1 / (x+3)^4 = infinity.
The books definition for definition 6 is, let f be a function defined on some open interval that contains a, except possibly at a itself, then lim x -> a f(x) = infinity means that for every positive number E there is a positive number delta such that 0 <|x-a| < delta then f(x) > E. I don't understand how to do this or what the definition even means really.
Can you please explain it step by step all the way to the answer? Thank you.
Certainly! Let's break down the steps to prove the given limit using Definition 6.
Step 1: Start with the definition of the limit.
The definition states that for a function f(x) and a limit L, the limit of f(x) as x approaches a is L. In this case, we want to prove that the limit of 1 / (x+3)^4 as x approaches -3 is infinity.
Step 2: Interpret the definition.
In this context, the limit approaching infinity means that as x gets closer and closer to -3 (but not equal to -3), the function 1 / (x+3)^4 becomes arbitrarily large. In other words, there is no upper bound for the value of f(x) as x approaches -3.
Step 3: Apply the definition.
According to Definition 6, we need to prove that for every positive number E, there exists a positive number delta such that if 0 < |x - (-3)| < delta, then f(x) > E.
Step 4: Prove the statement.
Let's consider a positive number E. We need to find a positive number delta such that if 0 < |x - (-3)| < delta, then f(x) > E. Since we want to prove the limit as x approaches -3, we can rewrite the inequality as 0 < |x + 3| < delta.
Now, let's manipulate the original function:
f(x) = 1 / (x+3)^4
Since we want to prove that f(x) approaches infinity, we can set the inequality f(x) > E as:
1 / (x+3)^4 > E
To simplify, we can take the reciprocal of both sides:
(x+3)^4 < 1/E
Taking the fourth root of both sides (and noting that the inequality sign does not change when both sides are raised to a positive even power), we get:
|x+3| < (1/E)^(1/4)
Thus, we have found a positive delta = (1/E)^(1/4) that satisfies the given condition for any positive E.
Step 5: Conclusion.
Since we have found a positive delta for every positive E, this proves that the limit of 1 / (x+3)^4 as x approaches -3 is indeed infinity.
To summarize, using Definition 6, we proved that for any positive E, there exists a positive delta such that if 0 < |x + 3| < delta, then 1 / (x+3)^4 > E. Consequently, the limit of 1 / (x+3)^4 as x approaches -3 is infinity.