# How do you evaluate the limit #cos((x^5+1)/(x^6+x^5+100))# as x approaches #-oo#?

The Reqd. Limit

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To evaluate the limit of cos((x^5+1)/(x^6+x^5+100)) as x approaches -oo, we can consider the highest power of x in the numerator and denominator. In this case, the highest power of x is x^6.

As x approaches -oo, x^6 approaches oo. Therefore, we can simplify the expression by dividing both the numerator and denominator by x^6.

After simplification, the expression becomes cos((1/x + 1/x^6 + 100/x^6)/(1 + 1/x + 100/x^6)).

As x approaches -oo, 1/x approaches 0 and 1/x^6 also approaches 0. Therefore, we can simplify the expression further by substituting these values.

After substitution, the expression becomes cos((0 + 0 + 0)/(1 + 0 + 0)).

Simplifying further, the expression becomes cos(0/1).

Since the cosine of 0 is equal to 1, the limit of cos((x^5+1)/(x^6+x^5+100)) as x approaches -oo is equal to 1.

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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.

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