Use the $\epsilon=\delta$ definition of limit to verify the limit statements given.

$$\lim _{(x, y) \rightarrow(0,0)}\left(2 x^{2}+3 y^{2}\right)=0$$

Use the definition of the limit of a function of two variables to verify the limit.

$$\lim _{(x, y) \rightarrow(1,-3)} y=-3$$

Assume that a function f is defined at x = c, and, with the aid of Definition mentioned, write down precisely what condition (involving $\epsilon$ and $\delta$ ) must be satisfied for f to be continuous at x=c. Explain why the condition $0<|x-c|<\delta$ can be replaced by $|x-c|<\delta$.

Young's modulus for steel is thousands of times greater than that for rubber. All other factors being equal, does this mean that steel stretches much more easily than rubber? Explain.

Use the formal definition of limit to verify the indicated limit. $\lim _{x \rightarrow-2} \frac{x^{2}+2 x}{x+2}=-2$

Use the $\epsilon=\delta$ definition of limit to verify the limit statements given.

$$\lim _{(x, y) \rightarrow(1,-1)} \frac{x^{2}-y^{2}}{x+y}=2$$

Find a value of $\delta>0$ such that the following statement holds: If $0<|x-2|<\delta$, then $\left|\frac{1}{x^2}-\frac{1}{4}\right|<\epsilon$ for $\epsilon=0.03$. Then find a value of $\delta$ that works for $\epsilon=0.01$.

Use the $\epsilon=\delta$ definition of limit to verify the limit statements given.

$$\lim _{(x, y) \rightarrow(0,0)}\left(x+y^{2}\right)=0$$

In the following exercise, use a graphing utility to make a table showing the values of $f(x, y)$ at the given points. Use the result to make a conjecture about the limit of $f(x, y)$ as $(x, y) \rightarrow(0,0)$. Determine whether the limit exists analytically and discuss the continuity of the function.

$$f(x, y)=\frac{y}{x^2+y^2}$$

Path: $y=0$

Points: $(1,0),(0.5,0),(0.1,0),(0.01,0),(0.001,0)$

Path: $y=x$

Points: $(1,1),(0.5,0.5),(0.1,0.1),(0.01,0.01),(0.001,0.001)$

Let k > 0 be a constant. Think of k as a constant function f(n) = k. Show that k is in Θ(1). k · 1 ≤ k ≤ k · 1 k · 1 ≤ k ≤ k · 0 k · 1 ≤ k ≤ 1 0 ≤ k ≤ k · 1 k · 0 ≤ k ≤ k · 1

Show that two nonvertical lines are parallel if and only if they have the same slope.

For the functions $f$ and $g$, find

$$f(g(x))$$

where $f(x)=e^x$ and $g(x)=x^2$.

Two students have scores with the same percentile, but for different administrations of the SAT. Does this mean that the students have the same score on the SAT? Explain your answer.

Use a computer algebra system to graph the function and find $\lim _{(x, y) \rightarrow(0,0)} f(x, y)$ (if it exists).

$$f(x, y)=\frac{10 x y}{2 x^2+3 y^2}$$