The concept of a limit

Reading

• Sections 2.1, 2.2

Practice problems

• Section 2.1: 1, 5, 25, 31
• Section 2.2: 1, 3, 9, 17, 21
• To turn in (Wednesday): 2.1 6, 8, 2.2 2, 22

Notes

Average and instantaneous rates of change

• Imagine someone’s position on the \(x\) axis as a function of time \(t\) is given by \(x=2t^2\).
• We can find their average velocity between two times by taking the ratio of the difference in the \(x\) positions over the difference in times.
• If we want to find out how fast the person is going at one specific time, their instantaneous velocity, that’s harder.
• Think of the value you see in the speedometer of a car. How is it computed?
• Idea: Measure very small time intervals. In fact, make them smaller and smaller.
• Practice: 2.1 1

Tangent lines

• Consider the graph of a function \(f\), and look at a point \((a, f(a))\) on it.
• What is the line that best describes the curve, near \(a\)?
• Idea: Start with the secant lines: Lines joining the point \(a\) and a nearby point.
• Look at the slopes of the secant lines as the nearby point gets closer and closer to \(a\).
• Practice: Try this out for \(f(x) = \sqrt{x}\) near \(x=1\).

Limits

• Limits express the idea of what happens to a function \(f(x)\) as we look at values of \(x\) very close to a specific number \(a\).
• Example: \(\frac{\sin x}{x}\) when \(x\) is near \(0\).
• Numerically: Try numbers \(x\) closer and closer to \(0\). See that the resulting values get closer and closer to 1.

• Definition of limit:

  We say that the limit of \(f(x)\) as \(x\) approaches \(a\) is \(L\), and we write: \[\lim_{x\to a}f(x) = L\] if the difference \(|f(x)-L|\) becomes arbitrarily small when \(x\) is sufficiently close (but not equal) to \(a\).

  In other words, the values of \(f(x)\) must get arbitrarily close to \(L\) when \(x\) is sufficiently close but not equal to \(a\).

• Simple examples: \(\displaystyle\lim_{x\to a} k = k\), \(\displaystyle\lim_{x\to a} x = a\)
• More complex example: \(\displaystyle\lim_{x\to 1} 2x+1 = 3\).

• Practice: 2.2 21

Infinite and one-sided limits

• Some times a function gets arbitrarily large/small as we approach the limit point.
• Example: \(\frac{1}{x}\) at \(x=0\).
• One-sided limits: \(\displaystyle\lim_{x\to 0^+}\frac{1}{x} = +\infty\)
• \(\displaystyle\lim_{x\to 0^-}\frac{1}{x} = -\infty\)