Differentiation 2: Limits

Overview:
  1. Gradient of a line passing through two points
  2. Limits
  3. Differentiation from first principles
  4. Differentiating expressions of the form \(kx^n\) with respect to \(x\)
  5. The gradient at a point on a curve
  6. Tangents and normals

 Part 2: Limits


Limits

Roughly speaking, a limit of a function is the value that function approaches as the input into the function approaches some value.

Consider the function \(\text{f}(x)=\dfrac{1}{x}\). As \(x\) gets larger, the value of \(\text{f}(x)\) decreases, getting closer and closer to 0, but never reaching or dropping below 0. We write that \(\displaystyle \lim_{x \to \infty} f(x) = 0\), and say this as follows: “the limit as \(x\) tends to infinity of \(\text{f}(x)\) is 0.”

This applet has four examples. In each case, the aim is to understand what happens to the value of \(\text{f}(x)\) as \(x\) increases. Use the yellow “+1” button to repeatedly increase the value of \(x\) by 1. Once you’ve done this a few times, consider the values of \(\text{f}(100)\text{, f}(1000)\text{, f}(1,000,000)\) etc. to get a sense of the limits reached by these functions.

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Compare the above examples with the function \(\text{f}(x)=x^2\). In this case, you can hopefully see that as \(x\) increases, so does the value of \(\text{f}(x)\). There is no limit in this case.

Exercise
Now have a go at finding the following:
  1. \(\displaystyle \lim_{x \to \infty} \frac{2}{x}\)
  2. \(\displaystyle \lim_{x \to \infty} \frac{3}{x}\)
  3. \(\displaystyle \lim_{x \to \infty} \left(\frac{3}{x}+7\right)\)
  4. \(\displaystyle \lim_{x \to \infty} \left(\frac{3}{x}-7\right)\)
  5. \(\displaystyle \lim_{x \to \infty} \frac{1}{x^2}\)
  6. \(\displaystyle \lim_{x \to -\infty} 2^{-x}\)
  7. \(\displaystyle \lim_{x \to -\infty} \left(2^{-x}+5\right)\)
  8. \(\displaystyle \lim_{n \to \infty} \frac{n}{n+1}\)
  9. \(\displaystyle \lim_{n \to \infty} \frac{n}{n+2}\)
  10. \(\displaystyle \lim_{n \to \infty} \frac{3n}{n+2}\)
  11. \(\displaystyle \lim_{n \to \infty} \frac{3n}{4n+2}\)
  12. \(\displaystyle \lim_{t \to 0} \frac{2t^2+7t}{t}\)
Answers
  1. \(\displaystyle \lim_{x \to \infty} \frac{2}{x}=0\)
  2. \(\displaystyle \lim_{x \to \infty} \frac{3}{x}=0\)
  3. \(\displaystyle \lim_{x \to \infty} \left(\frac{3}{x}+7\right)=7\)
  4. \(\displaystyle \lim_{x \to \infty} \left(\frac{3}{x}-7\right)=-7\)
  5. \(\displaystyle \lim_{x \to \infty} \frac{1}{x^2}=0\)
  6. \(\displaystyle \lim_{x \to -\infty} 2^{-x}=0\)
  7. \(\displaystyle \lim_{x \to -\infty} \left(2^{-x}+5\right)=5\)
  8. \(\displaystyle \lim_{n \to \infty} \frac{n}{n+1}=1\)
  9. \(\displaystyle \lim_{n \to \infty} \frac{n}{n+2}=1\)
  10. \(\displaystyle \lim_{n \to \infty} \frac{3n}{n+2}=3\)
  11. \(\displaystyle \lim_{n \to \infty} \frac{3n}{4n+2}=\frac{3}{4}\)
  12. \(\displaystyle \lim_{t \to 0} \frac{2t^2+7t}{t}=7\)