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Activity2.2Limits Laws

When proving the value of a limit we frequently rely upon laws that are easy to prove using the technical definitions of limit. These laws can be found in Appendix A. The first of these type laws are called replacement laws. Replacement laws allow us to replace limit expressions with the actual values of the limits.

Subsection2.2.1Exercises

The value of each of the following limits can be established using one of the replacement laws. Copy each limit expression onto your own paper, state the value of the limit (e.g. \(\lim\limits_{x\to9}5=5\)), and state the replacement law (by number) that establishes the value of the limit.

1

\(\lim\limits_{t\to\pi}t\)

2

\(\lim\limits_{x\to14}23\)

3

\(\lim\limits_{x\to14}x\)

4Applying Limit Laws

This exercise is fully worked out to serve as an example.

Find \(\lim_{x\to7}\left(4x^2+3\right)\text{.}\)

\begin{align*} \lim_{x\to7}\left(4x^2+3\right)&=\lim_{x\to7}\left(4x^2\right)+\lim_{x\to7}3&&\knowl{./knowl/lla1.html}{\text{Limit Law A1}}\\ &=4\lim_{x\to7}x^2+\lim_{x\to7}3&&\knowl{./knowl/lla3.html}{\text{Limit Law A3}}\\ &=4\left(\lim_{x\to7}x\right)^2+\lim_{x\to7}3&&\knowl{./knowl/lla6.html}{\text{Limit Law A6}}\\ &=4\cdot7^2+3&&\knowl{./knowl/llr1.html}{\text{Limit Law R1}}, \knowl{./knowl/llr2.html}{\text{R2}}\\ &=199 \end{align*}

The algebraic limit laws allow us to replace limit expressions with equivalent limit expressions. When applying limit laws our first goal is to come up with an expression in which every limit in the expression can be replaced with its value based upon one of the replacement laws. This process is shown in Exercise 2.2.1.4. Please note that all replacement laws are saved for the second to last step and that each replacement is explicitly shown. Please note also that each limit law used is referenced by number.

Use the limit laws to establish the value of each of the following limits. Make sure that you use the step-by-step, vertical format shown in Exercise 2.2.1.4. Make sure that you cite the limit laws used in each step. To help you get started, the steps necessary in Exercise 2.2.1.5 are outlined as:

  1. Apply Limit Law A6

  2. Apply Limit Law A1

  3. Apply Limit Law A3

  4. Apply Limit Law R1 and Limit Law R2

5

\(\lim\limits_{t\to4}\sqrt{6t+1}\)

6

\(\lim\limits_{y\to7}\dfrac{y+3}{y-\sqrt{y+9}}\)

7

\(\lim\limits_{x\to\pi}\left(x\cos(x)\right)\)