State the formal definition of a limit.

In the following exercise, find the value(s) of k that makes the function continuous over the given interval.

$$f(x)=\left\{\begin{aligned} \frac{x^{2}+3 x+2}{x+2}, & x \neq-2 \\ k, & x=-2 \end{aligned}\right.$$

Express each set in roster form. Set A is the set of odd natural numbers between 5 and 16.

A bank may compound interest over various lengths of time: yearly, half-yearly, quarterly, monthly, daily, and so on. If interest is compounded instantaneously, we can show that $I=\int_{0}^{T} P_{0} r e^{r t} d t$ where I is the interest accrued, $P_0$ is the initial investment, r is the rate of interest per annum as a decimal, and T is the period of the loan in years. a. Show that the amount of money in an account at time T is given by $P_{T}=P_{0} e^{r T}.$ b. How long will it take for an amount to double at a rate of 8% p.a.? c. A block of land was bought for $55 in 1940 and sold for$196000 in 2007 at the same time of the year. What rate of interest, compounded instantaneously, would produce this increase in the same time?

Suppose $f,g:\mathbb{R}\rightarrow\mathbb{R}$ are defined by $f(x)=x^2+x-1$ and $g(x)=3x+2$ Express, in simplest terms, $(f\circ g)(x)=(g \circ f)(x)$.

Evaluate the integral. integral (te^2ti+t/1-tj+1/(1-t^2)^1/2k)dt

For the following exercise, decide if the function continuous at the given point. If it is discontinuous, what type of discontinuity is it? $f(y)=\frac{\sin (\pi y)}{\tan (\pi y)}, \text { at } y=1$

Decide whether the function is continuous on the given interval. If not, define or redefine the function at one point to make it continuous everywhere on the given interval, or else explain why that cannot be done.

$$f(x)=\frac{x+4}{x-8} \text { on }[0,10]$$

For the following exercise, determine the point(s), if any, at which the function is discontinuous. Classify any discontinuity as jump, removable, infinite, or other. $f(x)=\frac{x}{x^{2}-x}$

In the following exercise, use direct substitution to show that the limit leads to the indeterminate form 0/0. Then, evaluate the limit. $\lim _{x \rightarrow 2} \frac{x-2}{x^{2}-2 x}$

In the following exercise, use the graph of the function y = f(x) shown here to find the values, if possible. Estimate when necessary. $\lim _{x \rightarrow 0^{-}} f(x)$

Verify each identity. $\sin ( 2 x - y ) \cos y + \cos ( 2 x - y ) \sin y = \sin 2 x$

A number to the third power increased by 9 A. 3n + 9 B. n^3 + 9 C. (3n)9 D. 9n^3

For the following exercise, consider the function $f(x)=-x^{2}+1$. Approximate the area of the region between the x-axis and the graph of f over the interval [-1, 1].