How do you graph and find the discontinuities of #y=(3x) /( 4-x^2)#?

Answer 1

For discontinuities you first look at the denominator.

That can be factorised into #(2-x)(2+x)#
So #x=2orx=-2# will make the denominator #=0# and are not allowed. Also the function will come to look like #-3/(4x)# as the #4# will make less and less difference as #x# grows larger, or:
#lim_(x-> oo)=0and lim_(x-> -oo)=0# graph{3x/(4-x^2) [-16.02, 16.01, -8.01, 8.01]}
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Answer 2

To graph and find the discontinuities of the function y = (3x) / (4 - x^2), follow these steps:

  1. Determine the domain of the function by identifying the values of x that make the denominator (4 - x^2) equal to zero. In this case, the denominator is a difference of squares, so set it equal to zero and solve for x: 4 - x^2 = 0 x^2 = 4 x = ±2

    Therefore, the domain of the function is all real numbers except x = ±2.

  2. Next, find the y-intercept by substituting x = 0 into the equation: y = (3(0)) / (4 - (0)^2) y = 0

    The y-intercept is (0, 0).

  3. To find the x-intercepts, set y = 0 and solve for x: 0 = (3x) / (4 - x^2) 3x = 0 x = 0

    The x-intercept is (0, 0).

  4. Determine the vertical asymptotes by analyzing the behavior of the function as x approaches the values that make the denominator zero. In this case, x = ±2 are the values that make the denominator zero. As x approaches ±2, the function approaches positive or negative infinity, respectively.

  5. Sketch the graph using the information obtained. The graph will have a vertical asymptote at x = -2 and x = 2, and it will pass through the point (0, 0).

Note: The graph will have a hole at (2, 3/4) and (-2, -3/4) due to the simplification of the function.

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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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