Pengertian
$ \lim_{x \to a} f(x) =
L $ adalah nilai f(x) dapat dibuat dekat ke L jika x
dekat ke-a
Pada penyelesaian soal
limit, hasil yang harus dihindari adalah
$\ \frac {0}{0} ; \
\frac { \infty}{\infty}; \ \infty - \infty; \ 0^0 ; \ \infty^0; \ 1^\infty$
A. Teorema Limit
a. Jika f(x) = c, maka $
\lim_{x \to a} f(x) = c $
b. $
\lim_{x \to a} f(x) \pm g(x) = \lim_{x \to a} f(x) \pm \lim_{x \to a} g(x)
$
c. $ \lim_{x \to a} f(x) . g(x) = \lim_{x \to a} f(x) .
\lim_{x \to a} g(x) $
d. $\lim_{x \to a} c . f(x) = c . \lim_{x \to a} f(x)$
e. $ \lim_{x \to a} \frac{f(x)}{g(x)} = \frac {\lim_{x \to
a} f(x)} { \lim_{x \to a} g(x)} $
f. $ \lim_{x \to a} f(x)^n = [\lim_{x \to a} f(x)]^n $
B. Limit Fungsi Trigonometri
1. $\lim_{x \to a} \frac{\sin ax}{bx} = \frac{a}{b} $
2. $\lim_{x \to a}
\frac{\cos ax}{bx} = \frac{a}{b} $
3. $\lim_{x \to a}
\frac{\tan ax}{bx} = \frac{a}{b} $
4. $\lim_{x \to a}
\frac{ \sin p(x - a)}{q(x - a)} = $ $\lim_{x \to a} \frac{p(x -
a)}{\sin q(x - a)} = \frac{p}{q} $
5. $\lim_{x \to a}
\frac{ \cos p(x - a)}{q(x - a)} = $ $\lim_{x \to a} \frac{p(x -
a)}{\cos q(x - a)} = \frac{p}{q} $
6. $\lim_{x \to a}
\frac{ \tan p(x - a)}{q(x - a)} = $ $\lim_{x \to a} \frac{p(x -
a)}{\tan q(x - a)} = \frac{p}{q} $
C. Limit Tak Hingga
1. $ \lim_{n \to \infty}
\ \frac{1}{n} = 0 $
2. $ \lim_{n \to \infty}
\ \sqrt[n]{p} = 1 ; p > 0$
3. $ \lim_{n \to
\infty}\ p^n = \infty ; | p | > 1 $
4. $ \lim_{n \to \infty}
\ p^n = 0 ; | p | < 1 $
5. $ \lim_{n \to \infty}
\ \frac{(1 + x)^n - 1}{x} = n $
6. $ \lim_{n \to \infty}
\ \frac{ax^p \ \ pm \ bx^{p-n} \ \pm ... \ \pm c }{dx^q \ \pm \ ex^{q-n} \ \pm
... \ \pm f} = \frac{a}{d} \ ; \ p = q $
7. $ \lim_{n \to \infty}
\ \frac{ax^p \ \ pm \ bx^{p-n} \ \pm ... \ \pm c }{dx^q \ \pm \ ex^{q-n} \ \pm
... \ \pm f} = \infty \ ; \ p > q $
8. $ \lim_{n \to \infty}
\ \frac{ax^p \ \ pm \ bx^{p-n} \ \pm ... \ \pm c }{dx^q \ \pm \ ex^{q-n} \ \pm
... \ \pm f} = 0 \ ; \ p < q $
D. Limit Aljabar
1. $ \lim_{n \to 0} \ \frac{x^m - 1}{px - 1} = m$
2. $ \lim_{n \to 0} \ \frac{arc \sin x}{x} = 1$
3. $ \lim_{n \to 0} \ \frac{arc \tan x}{x} = 1$
4. $ \lim_{n \to \infty} \ ( 1 + \frac{1}{x} )^{x} = e $
5. $ \lim_{n \to 0} \ ( 1 + x )^{\frac{1}{x}} = \frac{1}{e} $
6. $ \lim_{n \to 0} \ \frac{ln (1 + x)}{x} = 1 $
7. $ \lim_{n \to 0} \ \frac{log_a (1 + x)}{x}= \frac{1}{ln a}
$
8. $ \lim_{n \to 0} \ {a^x - 1}{x} = 1 $
E. Dalil L' Hospital
$ \lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a}
\frac{f('x)}{g'(x)} = \lim_{x \to a} \frac{f''(x)}{g''(x)} $
Syarat Kontinuitas pada x = a
1. f(a) harus ada nilainya
2. $ \lim_{x \to a} f(x) $ mempunyai
solusi
3. $ \lim_{x \to a} f(x) = f(a) $
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