6.116. lim _(xarrow infty )((x^2+x-1)/(x^2)-2x+5)^-2x

Question

Answer 1

To solve this limit, we can use the properties of exponential and logarithmic functions. Let's rewrite the expression as:

Now, let's take the natural logarithm of both sides:

Using the property of logarithms, we can simplify the expression:

Now, let's evaluate the limit:

As

approaches infinity, the terms

in the numerator and denominator will dominate. So, we can simplify the expression further:

As

approaches infinity, the fraction

approaches 0. Therefore, the natural logarithm term approaches 0:

So, the final answer is:

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