# How do you find horizontal asymptotes using limits?

Evaluate the limits as

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To find horizontal asymptotes using limits, you need to evaluate the limit of the function as x approaches positive or negative infinity.

- If the limit is a finite number, then there is a horizontal asymptote at that value.
- If the limit is positive or negative infinity, then there is no horizontal asymptote.
- If the limit does not exist, then there is no horizontal asymptote.

To find the limit, you can simplify the function and analyze the highest power of x in the numerator and denominator.

- If the highest power of x in the numerator is less than the highest power of x in the denominator, the horizontal asymptote is y = 0.
- If the highest power of x in the numerator is equal to the highest power of x in the denominator, divide the coefficients of the highest power terms to find the horizontal asymptote.
- If the highest power of x in the numerator is greater than the highest power of x in the denominator, there is no horizontal asymptote.

Remember to consider any restrictions on the domain of the function and to analyze the behavior of the function as x approaches positive or negative infinity.

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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.

- How do you find the limit of #(x^2 - 8) / (8x-16)# as x approaches #0^+#?
- What is the limit of #(ln(x+2)-ln(x+1))# as x approaches #oo#?
- How do you find the limit of #((16x)/(16x+3))^(4x)# as x approaches infinity?
- How do you prove that the limit of # (x²-9) / (x²+5x+6)=0 # as x approaches -3 using the epsilon delta proof?
- How do you find the limit of #(cos x - 1) / sin x^2# as x approaches 0?

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