How do you graph #f(x)=(x+2)/(x^2+4)# using holes, vertical and horizontal asymptotes, x and y intercepts?
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To graph the function ( f(x) = \frac{x + 2}{x^2 + 4} ), follow these steps:

Vertical Asymptotes: Find the vertical asymptotes by setting the denominator equal to zero and solving for ( x ). In this case, ( x^2 + 4 = 0 ) has no real solutions, so there are no vertical asymptotes.

Horizontal Asymptotes: Determine the horizontal asymptotes. As ( x ) approaches positive or negative infinity, the function approaches zero since the degree of the numerator is less than the degree of the denominator. Thus, the horizontal asymptote is ( y = 0 ).

Holes: Check if there are any holes in the graph by simplifying the function. In this case, there are no common factors between the numerator and denominator, so there are no holes.

xintercepts: Find the xintercepts by setting ( f(x) = 0 ) and solving for ( x ). In this case, ( \frac{x + 2}{x^2 + 4} = 0 ) has no real solutions, so there are no xintercepts.

yintercept: Find the yintercept by setting ( x = 0 ) and evaluating ( f(x) ). In this case, ( f(0) = \frac{0 + 2}{0^2 + 4} = \frac{2}{4} = \frac{1}{2} ), so the yintercept is at ( (0, \frac{1}{2}) ).
Plot the horizontal asymptote at ( y = 0 ) and the yintercept at ( (0, \frac{1}{2}) ). Since there are no vertical asymptotes or xintercepts, the graph does not intersect the xaxis. The graph approaches the xaxis but never touches it, as it gets closer to it.
This information is sufficient to sketch the graph of ( f(x) = \frac{x + 2}{x^2 + 4} ).
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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.
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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