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How do you find the limit of #(1+3/x)^(2x)# as x approaches negative infinity?

Answer 1

Luke Alvarez

#e^6#

#L = lim_{x \to - \infty} \ (1+3/x)^(2x)#

let #1/p = 3/x, \qquad p = x/3#

#\implies L = lim_{p \to - \infty} \ (1+1/p)^(6p) = lim_{p \to - \infty} \ ((1+1/p)^(p))^6#

From well known limit:

# \implies L = e^6#

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Answer 2

Scarlett Allison

To find the limit of (1+3/x)^(2x) as x approaches negative infinity, we can rewrite the expression using the natural logarithm function. Taking the natural logarithm of both sides, we get:

ln[(1+3/x)^(2x)]

Using the properties of logarithms, we can simplify this expression further:

2x * ln(1+3/x)

Now, as x approaches negative infinity, the term 3/x approaches 0. Therefore, we can rewrite the expression as:

2x * ln(1+0)

Since ln(1) is equal to 0, the expression simplifies to:

2x * 0 = 0

Thus, the limit of (1+3/x)^(2x) as x approaches negative infinity is 0.

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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.