Given the piecewise function defined below and its graph determine the following

Question

Given the piecewise function defined below and its graph determine the following.

a) Does the limit of this function exist as x approaches -2? Why or Why Not? If so, what is it?

b) Is this graph continuous at -2? Why or Why not? If it is not continuous what exists in the graph at x = -2?

c) Does the limit of this graph exist as x approaches zero? Why or Why Not?

d) Is this graph continuous at 4? Why or Why not?

e) What is the limit of this function as x approaches 4 from the left? How about the right? What exists in the graph at x = 4?

f) Does there appear to be a limit as x approaches positive infinity? If so, what is it? What exists in the graph as x approaches infinity?

g) g) Using set notation describe where this graph is continuous.

Explanation / Answer

a) limit of this function does not exist as x approaches -2 because value of function at x=-2(value at x=-2 is 2) is not equal to value of function as x approches -2(value is 9)

b)Graph is not continuous at -2 because value of function at x=-2(value at x=-2 is 2) is not equal to value of function as x approches -2(value is 9) .Removable discontinuity exists

c) limit of this graph exist as x approaches zero because value of function at x=0(value at x=0 is 0) is equal to value of function as x approches 0 from both sides(value is 0)

d)graph is not continuous at 4 because f(4) is undefined [denominator is x-4 and denominator cannot be 0]

e) limit of this function as x approaches 4 from the left is -infinite and limit of this function as x approaches 4 from the right is +infinite.Asymptotic discontinuity[also called essential discontinuity) exists here.

f)limit as x approaches positive infinity is 1.Removable discontinuity exists here.

g)[-infinte,-2)U(-2,4)U(4,infinite)

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