How do you graph #f(x)=2/(x^2+1)# using holes, vertical and horizontal asymptotes, x and y intercepts?
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To graph the function f(x) = 2/(x^2 + 1), we can analyze its holes, vertical and horizontal asymptotes, as well as the x and y intercepts.

Holes: The function has no holes.

Vertical Asymptotes: The denominator x^2 + 1 will never equal zero, so there are no vertical asymptotes.

Horizontal Asymptotes: As x approaches positive or negative infinity, the function approaches zero. Therefore, the horizontal asymptote is y = 0.

Xintercepts: To find the xintercepts, we set f(x) = 0 and solve for x. However, since the numerator is always 2 (nonzero), the function has no xintercepts.

Yintercept: To find the yintercept, we set x = 0 and evaluate f(0). Plugging in x = 0, we get f(0) = 2/(0^2 + 1) = 2/1 = 2. Therefore, the yintercept is (0, 2).
By considering these aspects, we can graph the function f(x) = 2/(x^2 + 1) accordingly.
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When evaluating a onesided 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 onesided 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 onesided 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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