Question

Find the limit of the function algebraically (with cancellation techniques and direct substitution, that is, since this is in indeterminate form):


`\lim_(x \to 0) (x^2+3)/(x^4)`
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It seems to have a dividing by zero issue from what I have managed to get out of the problem... That or I am having trouble discerning how to break it down into a manageable 0 issue. Help would be greatly appreciated, or at the very least a confirmation that I have the correct answer in that there is no limit.

Answer 1

At x=0, the numerator is positive and the denominator is zero. The form is not "indeterminate", but is "undefined". The limit is the same (+∞) whether approaching zero from the left or the right, so we say

the limit is +∞

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The case of "no limit" is reserved for those conditions in which the value you get approching from the left is different from that approaching from the right and/or is different from the defined value of the function at the point.

For example, 1/x has no limit at x=0 because it approaches -∞ from the left and +∞ from the right.