Limit Laws

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Graphs vs. Algebra

Example 1. Evaluate the following limits given the piecewise function

\[ f(x) = \left\{ \begin{array}{ll} -x, & x < -2\\ \cos(3x), & -2\le x \le 1\\ \sqrt{x}, & x>1 \end{array}\right. \]

• Evaluate \(\displaystyle\lim_{x\to (-2)^-} f(x)\).
• Evaluate \(\displaystyle\lim_{x\to (-2)^+} f(x)\).
• Evaluate \(\displaystyle\lim_{x\to 1^-} f(x)\).
• Evaluate \(\displaystyle\lim_{x\to 1^+} f(x)\).

Practice Problems.

Evaluate the following limits based on the given graph.

• \(\displaystyle\lim_{x\to (-1)^-} f(x)=\)
• \(\displaystyle\lim_{x\to (-1)^+} f(x)=\)
• \(\displaystyle\lim_{x\to (-1)} f(x)=\)
• \(f(-1)=\)

• \(\displaystyle\lim_{x\to 0^-} f(x)=\)
• \(\displaystyle\lim_{x\to 0^+} f(x)=\)
• \(\displaystyle\lim_{x\to 0} f(x)=\)
• \(f(0)=\)

• \(\displaystyle\lim_{x\to 1^-} f(x)=\)
• \(\displaystyle\lim_{x\to 1^+} f(x)=\)
• \(\displaystyle\lim_{x\to 1} f(x)=\)
• \(f(1)=\)

Are you able to make the connection between reading limits on a graph and evaluating limits by algebra?

Limit Laws

If \(\displaystyle\lim_{x \to a} f(x)\) and \(\displaystyle\lim_{x \to a} g(x)\) exist, then

• \(\displaystyle \lim_{x\to a} \big(f(x) + g(x)\big) = \lim_{x\to a} f(x) + \displaystyle \lim_{x\to a} g(x)\)
• \(\displaystyle \lim_{x\to a} \big(f(x) - g(x)\big) = \lim_{x\to a} f(x) - \displaystyle \lim_{x\to a} g(x)\)
• \(\displaystyle \lim_{x\to a} c\cdot f(x) = c\cdot\lim_{x\to a} f(x)\)
• \(\displaystyle \lim_{x\to a} \big(f(x) \cdot g(x)\big) = \lim_{x\to a} f(x) \cdot \displaystyle \lim_{x\to a} g(x)\)
• \(\displaystyle \lim_{x\to a} \frac{f(x)}{g(x)} = \frac{\displaystyle\lim_{x\to a} f(x)}{\displaystyle \lim_{x\to a} g(x)}\) given that \(\displaystyle \lim_{x\to a} g(x)\ne 0\)
• \(\displaystyle \lim_{x\to a} \big(f(x)\big)^n = \big(\lim_{x\to a} f(x)\big)^n\)
  
• \(\displaystyle \lim_{x\to a} \sqrt[m]{f(x)} = \sqrt[m]{\lim_{x\to a} f(x)}\)

Example 2. Which of the above limit laws are used to evaluate the following limit? \[\displaystyle\lim_{x\to 0^+} \dfrac{3x^2\cos x - 1}{\sqrt{x} + 2}\]

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Essential Algebraic Skills

At first glance, do you see why the limit laws fail in evaluating all of the following examples without any algebraic manipulation?

\(\frac{0}{0}\) is an interesting form that we will revisit at a later time with a more fancy tool. I promise.

• Factor a quadratic/cubic polynomial.

Example 3. \(\displaystyle\lim_{x\to 3} \dfrac{x - 3}{x^2 - 9}\)

Example 4. \(\displaystyle\lim_{u\to 2} \dfrac{u^3 - 8}{u^2 - 5u + 6}\)

• Piecewise from an absolute value.

Example 5. \(\displaystyle\lim_{w\to 3^+} \dfrac{w^2 + w -12}{|w - 3|}\)

Example 6. \(\displaystyle\lim_{s\to 2^+} \dfrac{s^2 - 6s +8}{|2 - s|}\)

• Rationalize a radical.

Example 7. \(\displaystyle\lim_{x\to 0} \dfrac{\sqrt{2-x} - \sqrt{2 + x}}{x}\)

Example 8. \(\displaystyle\lim_{p\to 1} \dfrac{\sqrt{p} - 1}{\sqrt{p + 3} - 2}\)

• Fractions over fractions.

Example 9. \(\displaystyle\lim_{x\to 3} \dfrac{\frac 1 3 - \frac 1 x}{x - 3}\)

Example 10. \(\displaystyle\lim_{h\to 0} \dfrac{\frac{1}{(x+h)^2} - \frac{1}{x^2}}{h}\)

Verify your answers graphically.

Squeeze Theorem

Suppose \(f(x) \le g(x) \le h(x)\) on a open interval containing \(x=a\), except perhaps at \(a\) where the functions may not even be defined. If \[\lim_{x \to a} f(x) = \lim_{x \to a} h(x)=L,\] for some real number \(L\), then \[\lim_{x \to a} g(x)=L.\]

A graphic example.

Example 11. Evaluate \(\displaystyle \lim_{x\to 0} x^2\sin \left(\frac 1 x\right)\)