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MA441- LEC 5 - 1.6 Calculating limits using the limit Laws (B)

By KWANG

Goal

  • Limit Laws and their restrictions

Theorem

\[ \lim_{x\to a} f(x)=L \Leftrightarrow \lim_{x\to a^-} f(x)=L=\lim_{x\to a^+} f(x) \]

Example 6.

\[ \lim_{x\to 0} |x| \]

Example 7.

\[ \lim_{x\to 0} \frac{|x|}{x} \]

Example 8.

\[ \lim_{x\to 0} \frac{x^2}{|x|} \]

Example 9.

\[ \lim_{x\to 0} \sqrt{x}=\]

Example 10.

\[ f(x)=\left\{ \begin{array}{ll} \sqrt{x-4} &, x>4\\ 8-2x &, x<4 \end{array} \right. \]

  1. Find \[ \lim_{x\to 4} f(x)\]
  2. Graph \( y=f(x) \).

Theorem( inequality of limit)

Assume

  1. \(f(x)\leq g(x)\) when \(x\) is near \(a\) (\(x\neq a\))
  2. \(\lim_{x\to a} f(x)\) and \(\lim_{x\to a} g(x)\) exists.

Then

\[ \lim_{x\to a} f(x)\leq \lim_{x\to a} g(x)\]

Remark

Strict inequality does not hold for the limits.


Ex. For \(x>0\), \(\frac{1}{x}>0\) But \( \lim_{x\to \infty} \frac{1}{x}=0\)

(The squeeze Theorem)

Assume

  1. \(f(x)\leq g(x)\leq h(x)\) when \(x\) is near \(a\)(\(x\neq a\))
  2. \(\displaystyle \lim_{x\to a} f(x)=\lim_{x\to a} h(x)=L\)

Then


\[ \lim_{x\to a} g(x)=L \] picw300

Example 11 Using the squeeze lemma, find

\[ \lim_{x\to a} \left(x^2\cdot\sin(\frac{1}{x}) \right) \]

We cannot say

\[ \lim_{x\to a} x^2\cdot\sin(\frac{1}{x})=(\lim_{x\to a} x^2)\cdot\lim_{x\to a}(\sin(\frac{1}{x})) \] Why not?

picw300