Katelyn C.

Formal Definition of Limit Given the limit

$\lim _{x \rightarrow 2}(2 x+1)=5$
use a sketch to show the meaning of the phrase
$$" 0<|x-2|<0.25$ implies $|(2 x+1)-5|<0.5"$$

Estimating a Limit Numerically In Exereises $11-18$ ,create a table of values for the function and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result.

$$\lim _{x \rightarrow-3} \frac{x^{3}+27}{x+3}$$

Finding a $\delta$ for a Given $\varepsilon$ Repeat Exercise 37 for $\varepsilon=0.05,0.01,$ and $0.005 .$ What happens to the value of $\delta$ as the value of $\varepsilon$ gets smaller?

Finding a $\delta$ for a Given $\varepsilon$ In Exercises $39-44,$ find the limit $L .$ Then find $\delta$ such that $|f(x)-L|<\varepsilon$ whenever $0<|x-c|<\delta$ for (a $)$ $\varepsilon=0.01$ and $(b) \varepsilon=0.005 .$

$$\lim _{x \rightarrow 6}\left(6-\frac{x}{3}\right)$$

Finding a $\delta$ for a Given $\varepsilon$ In Exercises $39-44,$ find the limit $L .$ Then find $\delta$ such that $|f(x)-L|<\varepsilon$ whenever $0<|x-c|<\delta$ for (a $)$ $\varepsilon=0.01$ and $(b) \varepsilon=0.005 .$

$$\lim _{x \rightarrow 4}\left(x^{2}+6\right)$$

Finding a $\delta$ for a Given $\varepsilon$ In Exercises $39-44,$ find the limit $L .$ Then find $\delta$ such that $|f(x)-L|<\varepsilon$ whenever $0<|x-c|<\delta$ for (a $)$ $\varepsilon=0.01$ and $(b) \varepsilon=0.005 .$
$$\lim _{x \rightarrow 3} x^{2}$$