Finding Limits - Teaching Calculus

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Ways to find limits (summary):

  1. If the function is continuous at the value x approaches, then substitute that value and the number you get will be the limit.
  2. If the function is not continuous at the value x approaches, then
    1. If you get something that is not zero divided by zero, the limit does not exist (DNE) or equals infinity (see below).
    2. If you get \frac{0}{0} or \frac{\infty }{\infty } the limit may exist. Simplify by factoring, or using different trig functions. Later in the year a method called L’Hôpital’s Rule can often be used in this situation.
  3. Dominance is a quick way of finding many limits. Exponentials dominate, polynomials, polynomials dominate logarithms, higher powers dominate lower powers. The next post will give some hints about dominance.

Infinity is not a number, but it often is used as if it were. When we say a limit is infinity, what we mean is that the function increases without bound, or there is some x-value that will make the expression larger than any number you choose. Writing things like \infty -\infty =0,\frac{\infty }{\infty }=1,\infty +\infty =2\infty  are common mistakes.

DNE or Infinity?  \displaystyle \underset{x\to 3}{\mathop{\lim }}\,\frac{1}{{{\left( x-3 \right)}^{2}}} does not exist, and DNE is a correct answer. However, it is a bit better to say the limit is (equals) infinity, indicating that the expression gets larger without bound as x approaches 3. Both answers will get credit on an AP exam.  \displaystyle \underset{x\to 3}{\mathop{\lim }}\,\frac{1}{x-3} DNE since the one-sided limits (from the left and from the right) are different.  Only DNE gets credit here.

Take a look at this AP question 1998 AB-2: In (a) students found that \displaystyle \underset{x\to \infty }{\mathop{\lim }}\,2x{{e}^{2x}}=\infty \text{ or }DNE, in (b) they found the minimum value of 2x{{e}^{2x}}  is -{{e}^{-1}} and in (c) they had to state the range of the function is [-{{e}^{-1}},\infty )\text{ or }x>-{{e}^{-1}}. Thus making the students show they knew that this kind of DNE is the kind where the value increases without bound.

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Tag » What Does Dne Mean In Math