함수의 극한과 연속 전체 문제

For the following exercises, construct a function $f(x)$ that has the given asymptotes.

$x=0$

For $f(x)=\frac{P(x)}{Q(x)}$ to have an asymptote at $y=2$ then the polynomials $P(x)$ and $Q(x)$ must have what relation?

For $f(x)=\frac{P(x)}{Q(x)}$ to have an asymptote at $x=0,$ then the polynomials $P(x)$ and $Q(x).$ must have what relation?

If ${f}^{'}(x)$ has asymptotes at $y=3$ and $x=1,$ then $f(x)$ has what asymptotes?

Both $f(x)=\frac{1}{(x-1)}$ and $g(x)=\frac{1}{{(x-1)}^{2}}$ have asymptotes at $x=1$ and $y=0.$ What is the most obvious difference between these two functions?

In Exercises 7–16, approximate the given limits both numerically and graphically.

$\displaystyle \lim_{x\to 1} x^2+3x-5$

In Exercises 7–16, approximate the given limits both numerically and graphically.

$\displaystyle \lim_{x\to 0} x^3-3x^2+x-5$

In Exercises 7–16, approximate the given limits both numerically and graphically.

$\displaystyle \lim_{x\to 0} \frac{x+1}{x^2+3x}$

In Exercises 7–16, approximate the given limits both numerically and graphically.

$\displaystyle \lim_{x\to 3} \frac{x^2-2x-3}{x^2-4x+3}$

In Exercises 7–16, approximate the given limits both numerically and graphically.

$\displaystyle \lim_{x\to -1} \frac{x^2+8x+7}{x^2+6x+5}$

In Exercises 7–16, approximate the given limits both numerically and graphically.

$\displaystyle \lim_{x\to 2} \frac{x^2+7x+10}{x^2-4x+4}$

In Exercises 7–16, approximate the given limits both numerically and graphically.

$\displaystyle \lim_{x\to 2} f(x)$, where

$f(x) = \left\{\begin{array}{cl} x+2 & x\leq 2 \\ 3x-5 & x>2 \end{array}\right.$.

In Exercises 7–16, approximate the given limits both numerically and graphically.

$\displaystyle \lim_{x\to 3} f(x)$, where

$f(x) = \left\{\begin{array}{cl} x^2-x+1 & x\leq 3 \\ 2x+1 & x>3 \end{array}\right.$.

In Exercises 7–16, approximate the given limits both numerically and graphically.

$\displaystyle \lim_{x\to 0} f(x)$, where

$f(x) = \left\{\begin{array}{cl} \cos x & x\leq 0 \\ x^2+3x+1 & x>0 \end{array}\right.$.

In Exercises 7–16, approximate the given limits both numerically and graphically.

$\displaystyle \lim_{x\to \pi/2} f(x)$, where

$f(x) = \left\{\begin{array}{cl} \sin x & x\leq \pi/2 \\ \cos x & x>\pi/2 \end{array}\right.$.

In Exercises 17–24, a function $f$ and a value $a$ are given. Approximate the limit of the difference quotient, $\displaystyle \lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$, using $h = \pm 0.1, \pm 0.01$.

$f(x) = -7x+2$, $a=3$

In Exercises 17–24, a function $f$ and a value $a$ are given. Approximate the limit of the difference quotient, $\displaystyle \lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$, using $h = \pm 0.1, \pm 0.01$.

$f(x) = 9x+0.06$, $a=-1$

In Exercises 17–24, a function $f$ and a value $a$ are given. Approximate the limit of the difference quotient, $\displaystyle \lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$, using $h = \pm 0.1, \pm 0.01$.

$f(x) = x^2+3x-7$, $a=1$

In Exercises 17–24, a function $f$ and a value $a$ are given. Approximate the limit of the difference quotient, $\displaystyle \lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$, using $h = \pm 0.1, \pm 0.01$.

$\displaystyle f(x) = \frac{1}{x+1}$, $a=2$

In Exercises 17–24, a function $f$ and a value $a$ are given. Approximate the limit of the difference quotient, $\displaystyle \lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$, using $h = \pm 0.1, \pm 0.01$.

$\displaystyle f(x) = -4x^2+5x-1$, $a=-3$