Prove the statement using the epsilon, delta definition of limit.

$$\lim _{x \rightarrow 3} \frac{x^2+x-12}{x-3}=7$$

Using the epsilon-delta definition of a limit we can show that $\lim_{x\to1} 3x+2 = 5$. True or False?

In your own words, state the geometric interpretations of epsilon and delta in the definition of

limit of f(x,y) as (x,y) approaches to (x sub 0, y sub 0) = L

given in the definition in this section.

Let $f(x, y)=x^{2}+3 y^{2}$ Use the $\epsilon-\delta$ definition of limits to show that $\lim _{(x, y) \rightarrow(0,0)} f (x, y) = 0$

Write a definition for $\lim\limits_{n \rightarrow \infty}s_{n} = L$ similar to the $\epsilon,\delta$ definition for $\lim\limits_{x \rightarrow a}f(x) = L$ in Section $1.8.$ Instread of $\delta,$ you will need $N$, a value of $n.$

Use the precise definition of a limit to prove the following limits. Specify a relationship between ${\varepsilon}$ and $\delta$ that guarantees the limit exists. $\lim _{x \rightarrow 3} x^{3}=27$

Prove the statement using the ε, 𝛿 definition of a limit lim x2 - 2x - 8 / x - 4 = 6 x-->1

For the limit

$$\displaystyle\lim_{x \to 0}\dfrac{e^x - 1}{x} = 1$$

illustrate the appropriate definition by finding values of $\delta$ that correspond to $\epsilon = 0.5$ and $\epsilon = 0.1$.

Prove the statement using the ε, 𝛿 definition of a limit and illustrate with a diagram like Figure 9. lim (2x - 5) = 3 x-->4

Prove the statement using ε, 𝛿 function definition of a limit. lim (x2 + 2x - 7) = 1 x-->2

Prove that if the limit of f(x) as x approaches c exists, then the limit must be unique. [Hint: Let lim_(x→c) f(x) = L_1and lim_(x→c) f(x) = L_2 and prove that L_1 = L_2.]

State the formal definition of a limit.

Prove the statement using ε, 𝛿 function definition of a limit. lim 2 + 4x / 3 = 2 x--->1

Definition of Limit I.

a) Write the formal definition of limit.

b) The graph of $f(x)=3 x-7$ is shown in figure. Show that f(4)=5.

c) You can keep f(x) close to 5 just by keeping x close to 4. How close to 4 must you keep x in order for f(x) to stay within 0.6 unit of 5 ?

d) The 0.6 in part c is a value of epsilon in the definition of limit. The answer to part c is a value of delta. Sketch how epsilon and delta are related to the graph.

e) Pick a value of x that is within delta units of 4 but not equal to 4 . Show that f(x) really is within epsilon units of 5 .

f) The number 5 fits the definition of limit because you could find a value for delta no matter how small epsilon is. Show that you understand the meaning of this statement by finding a value of delta if epsilon is 0.00012. Tell how you found this value of delta.