How do you graph #f(x)=2/(x^2+1)# using holes, vertical and horizontal asymptotes, x and y intercepts?

Answer 1

#f(x)# has an absolute maximum of 2 at #x=0#
#f(x) -> 0# as #x-> +- oo#

#f(x) = 2/(x^2+1)#
Since #x^2 + 1 > 0 forall x in RR# there exists no holes in #f(x)#
Also, #lim_"x-> +-oo" f(x) = 0#
#f'(x) = (-4x)/(x^2+1)^2#
For a maximum or minimum value; #f'(x) = 0#
#:. (-4x)/(x^2+1)^2 = 0 -> x=0#
#f(0) = 2/(0+1) = 2#
Since #f''(0) < 0# #f(0) = 2# is a maximum of #f(x)#
The critical points of #f(x)# can be seen on the graph below: graph{2/(x^2+1) [-5.55, 5.55, -2.772, 2.778]}
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Answer 2

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.

  1. Holes: The function has no holes.

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

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

  4. X-intercepts: To find the x-intercepts, we set f(x) = 0 and solve for x. However, since the numerator is always 2 (non-zero), the function has no x-intercepts.

  5. Y-intercept: To find the y-intercept, 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 y-intercept is (0, 2).

By considering these aspects, we can graph the function f(x) = 2/(x^2 + 1) accordingly.

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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.

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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