함수의 극한의 뜻

Use the value in the preceding exercise to find an equation of the tangent line at point P.

Use the value in the preceding exercise to find an equation of the tangent line at point P.

Use the value in the preceding exercise to find an equation of the tangent line at point P.

What do your results in the preceding exercise indicate about the two-sided limit $\lim_{x\to 1}f(x)?$ Explain your response.

To which mathematical constant does the limit in the preceding exercise appear to be getting closer?

Use the preceding two exercises to conjecture (guess) the value of the following limit: $\lim_{x\to 0}\frac{\sin ax}{x}$ for a, a positive real value.

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 $h = \pm 0.1, \pm 0.01$.

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