How do you graph and find the discontinuities of #(x^225)/(x^2+5x)#?
We first factorise.
graph{15/x [16.02, 16, 8, 8.02]}
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To graph the function (x^225)/(x^2+5x) and find its discontinuities, follow these steps:

Factor the numerator and denominator: (x^225) can be factored as (x+5)(x5) (x^2+5x) can be factored as x(x+5)

Simplify the function by canceling out common factors: (x+5)(x5) / x(x+5)

Identify any values of x that would make the denominator equal to zero: x = 0 and x = 5 are the values that make the denominator zero.

Determine the vertical asymptotes: Vertical asymptotes occur at the values of x that make the denominator zero, so in this case, x = 0 and x = 5.

Plot the points on the graph: Plot the points (0, undefined) and (5, undefined) to represent the vertical asymptotes.

Determine the horizontal asymptote: To find the horizontal asymptote, compare the degrees of the numerator and denominator. In this case, both have a degree of 2, so the horizontal asymptote is y = 1.

Plot the horizontal asymptote on the graph.

Determine any other points of interest: You can choose additional xvalues to evaluate the function and plot corresponding points on the graph.

Connect the points smoothly to create the graph.

The discontinuities of the function are at x = 0 and x = 5, where the function is undefined.
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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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