함수의 극한과 연속 전체 문제

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

$\displaystyle f(x) =\sin x$, $a=\pi$

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

$\displaystyle f(x) =\cos x$, $a=\pi$

Explain in your own words, without using $\epsilon$-$\delta$ formality, why $\displaystyle \lim_{x\to c} b = b$.

Explain in your own words, without using $\epsilon$-$\delta$ formality, why $\displaystyle \lim_{x\to c} x = c$.

What does the text mean when it says that certain functions' “behavior is `nice' in terms of limits”? What, in particular, is “nice”?

Sketch a graph that visually demonstrates the Squeeze Theorem.

You are given the following information:

(a) $\displaystyle \lim_{x\to 1} f(x) = 0$ (b) $\displaystyle \lim_{x\to 1} g(x) = 0$ (c) $\displaystyle \lim_{x\to 1} f(x)/g(x) = 2$

What can be said about the relative sizes of $f(x)$ and $g(x)$ as $x$ approaches 1?

In Exercises 7–14

, use the following information to evaluate the given limit, when possible. If it is not possible to determine the limit, state why not.

• $\displaystyle \lim_{x\to9}f(x) = 6$, $\displaystyle \lim_{x\to6} f(x) = 9$, $f(9)=6$
• $\displaystyle \lim_{x\to9}g(x) = 3$, $\displaystyle \lim_{x\to6} g(x) = 3$, $g(6)=9$

$\displaystyle \lim_{x\to9}(f(x)+g(x))$

In Exercises 7–14

, use the following information to evaluate the given limit, when possible. If it is not possible to determine the limit, state why not.

• $\displaystyle \lim_{x\to9}f(x) = 6$, $\displaystyle \lim_{x\to6} f(x) = 9$, $f(9)=6$
• $\displaystyle \lim_{x\to9}g(x) = 3$, $\displaystyle \lim_{x\to6} g(x) = 3$, $g(6)=9$

$\displaystyle \lim_{x\to9}(3f(x)/g(x))$

In Exercises 7–14

, use the following information to evaluate the given limit, when possible. If it is not possible to determine the limit, state why not.

• $\displaystyle \lim_{x\to9}f(x) = 6$, $\displaystyle \lim_{x\to6} f(x) = 9$, $f(9)=6$
• $\displaystyle \lim_{x\to9}g(x) = 3$, $\displaystyle \lim_{x\to6} g(x) = 3$, $g(6)=9$

$\displaystyle \lim_{x\to9}\left(\frac{f(x)-2g(x)}{g(x)}\right)$

In Exercises 7–14

, use the following information to evaluate the given limit, when possible. If it is not possible to determine the limit, state why not.

• $\displaystyle \lim_{x\to9}f(x) = 6$, $\displaystyle \lim_{x\to6} f(x) = 9$, $f(9)=6$
• $\displaystyle \lim_{x\to9}g(x) = 3$, $\displaystyle \lim_{x\to6} g(x) = 3$, $g(6)=9$

$\displaystyle \lim_{x\to6}\left(\frac{f(x)}{3-g(x)}\right)$

In Exercises 7–14

, use the following information to evaluate the given limit, when possible. If it is not possible to determine the limit, state why not.

• $\displaystyle \lim_{x\to9}f(x) = 6$, $\displaystyle \lim_{x\to6} f(x) = 9$, $f(9)=6$
• $\displaystyle \lim_{x\to9}g(x) = 3$, $\displaystyle \lim_{x\to6} g(x) = 3$, $g(6)=9$

$\displaystyle \lim_{x\to9}g\big(f(x)\big)$

In Exercises 7–14

, use the following information to evaluate the given limit, when possible. If it is not possible to determine the limit, state why not.

• $\displaystyle \lim_{x\to9}f(x) = 6$, $\displaystyle \lim_{x\to6} f(x) = 9$, $f(9)=6$
• $\displaystyle \lim_{x\to9}g(x) = 3$, $\displaystyle \lim_{x\to6} g(x) = 3$, $g(6)=9$

$\displaystyle \lim_{x\to6}f\big(g(x)\big)$

In Exercises 7–14

, use the following information to evaluate the given limit, when possible. If it is not possible to determine the limit, state why not.

• $\displaystyle \lim_{x\to9}f(x) = 6$, $\displaystyle \lim_{x\to6} f(x) = 9$, $f(9)=6$
• $\displaystyle \lim_{x\to9}g(x) = 3$, $\displaystyle \lim_{x\to6} g(x) = 3$, $g(6)=9$

$\displaystyle \lim_{x\to6}g\big(f(f(x))\big)$

In Exercises 7–14

, use the following information to evaluate the given limit, when possible. If it is not possible to determine the limit, state why not.

• $\displaystyle \lim_{x\to9}f(x) = 6$, $\displaystyle \lim_{x\to6} f(x) = 9$, $f(9)=6$
• $\displaystyle \lim_{x\to9}g(x) = 3$, $\displaystyle \lim_{x\to6} g(x) = 3$, $g(6)=9$

$\displaystyle \lim_{x\to6}f(x)g(x)-f\,^2(x)+g^2(x)$

In Exercises 15–18, use the following information to evaluate the given limit, when possible. If it is not possible to determine the limit, state why not.

• $\displaystyle \lim_{x\to1}f(x) = 2$, $\displaystyle \lim_{x\to10} f(x) = 1$, $f(1)=1/5$
• $\displaystyle \lim_{x\to1}g(x) = 0$, $\displaystyle \lim_{x\to10} g(x) = \pi$, $g(10)=\pi$

$\displaystyle \lim_{x\to1}f(x)^{g(x)}$

In Exercises 15–18, use the following information to evaluate the given limit, when possible. If it is not possible to determine the limit, state why not.

• $\displaystyle \lim_{x\to1}f(x) = 2$, $\displaystyle \lim_{x\to10} f(x) = 1$, $f(1)=1/5$
• $\displaystyle \lim_{x\to1}g(x) = 0$, $\displaystyle \lim_{x\to10} g(x) = \pi$, $g(10)=\pi$

$\displaystyle \lim_{x\to10}\cos \big(g(x)\big)$

In Exercises 15–18, use the following information to evaluate the given limit, when possible. If it is not possible to determine the limit, state why not.

• $\displaystyle \lim_{x\to1}f(x) = 2$, $\displaystyle \lim_{x\to10} f(x) = 1$, $f(1)=1/5$
• $\displaystyle \lim_{x\to1}g(x) = 0$, $\displaystyle \lim_{x\to10} g(x) = \pi$, $g(10)=\pi$

$\displaystyle \lim_{x\to1}f(x)g(x)$

In Exercises 15–18, use the following information to evaluate the given limit, when possible. If it is not possible to determine the limit, state why not.

• $\displaystyle \lim_{x\to1}f(x) = 2$, $\displaystyle \lim_{x\to10} f(x) = 1$, $f(1)=1/5$
• $\displaystyle \lim_{x\to1}g(x) = 0$, $\displaystyle \lim_{x\to10} g(x) = \pi$, $g(10)=\pi$

$\displaystyle \lim_{x\to1}g\big(5f(x)\big)$

In Exercises 19–34, evaluate the given limit.

$\displaystyle \lim_{x\to3}x^2-3x+7$