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What is the limit of #(x^2 + x + 4)/(x^3 - 2x^2 + 7)# as x approaches a and when does the limit exist?

Answer 1

Let #f(x) = (x^2 + x + 4)/(x^3 - 2x^2 + 7)#

#lim_(xrarra) f(x) = f(a)# if #x=a# does not make the denominator #0#,

That is: if #a# is not a solution to #x^3 - 2x^2 + 7 = 0#. (There is one negative and no positive solutions. Though it takes some algebra to see that.)

Additionally, if the numerator and denominator have any common real zeros, (say #c#) then the limit as #xrarrc# will exist, but we'll have to simplify the expression to find the limit.

#x^2+x+4# has no real zeros and is not a factor of #x^3- 2x^2 + 7#, so that case will not arise.

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Answer 2

Cameron Alexander

The limit of (x^2 + x + 4)/(x^3 - 2x^2 + 7) as x approaches a is determined by evaluating the expression when x is very close to a. The limit exists if the expression approaches a specific value as x gets arbitrarily close to a. To determine if the limit exists, we need to check if the denominator becomes zero at x = a. If the denominator does not equal zero at x = a, then the limit exists.

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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.