Where is the function #f(x) = x^4-1/x^2# discontinuous?

Answer 1

x=0

It has to be considered that function gets a real and finite value as x takes up a value in the interval #(-oo, +oo)#. For the given function it is obvious that at x=0, the function becomes infinite, It is at this point, the function is said to be not continuous.
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Answer 2

This function is continuous on the whole of its domain, but its domain excludes the point #x=0#

#f(x) = x^4-1/(x^2)# is defined for all #x in (-oo, 0) uu (0, oo)#

and constant throughout this entire domain:

If #a != 0# then #lim_(x->a) f(x) = f(a)#
To see this, let #a != 0# and #abs(epsilon) < abs(a)#
#f(a+epsilon) - f(a)#
#=((a+epsilon)^4-1/((a+epsilon)^2))-(a^4-1/(a^2))#
#=((a+epsilon)^4 - a^4)+(1/(a^2)-1/((a+epsilon)^2))#
#=(4a^3epsilon+6a^2epsilon^2+4a epsilon^3+epsilon^4)+(((a+epsilon)^2 - a^2)/(a^2(a+epsilon)^2))#
#=(4a^3epsilon+6a^2epsilon^2+4a epsilon^3+epsilon^4)+((2a epsilon + epsilon^2)/(a^2(a+epsilon)^2))#
#=epsilon(4a^3+6a^2epsilon+4a epsilon^2+epsilon^3+(2a+epsilon)/(a^2(a+epsilon)^2))#
#->0# as #epsilon -> 0#
Note also that #f(0)# is not defined so #0# is not in the domain of #f#.
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Answer 3

The function f(x) = x^4 - 1/x^2 is discontinuous at x = 0 due to the presence of a vertical asymptote at that point.

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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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