Yfirlitsverkefni

lim𝑥→−1+⁡𝑓⁡(𝑥)$\underset{x\to {\mathrm{-1}}^{+}}{\lim }f(x)$

lim𝑥→−1−⁡𝑓⁡(𝑥)$\underset{x\to {\mathrm{-1}}^{-}}{\lim }f(x)$

lim𝑥→−1⁡𝑓⁡(𝑥)$\underset{x\to -1}{\lim }f(x)$

lim𝑥→3⁡𝑓⁡(𝑥)$\underset{x\to 3}{\lim }f(x)$

Fyrir hvaða gildi $x$ er fallið ósamfellt? Hvaða samfelldniskilyrði er ekki uppfyllt?

Notaðu töflu 1 til að áætla $\underset{x\to 0}{\lim }f(x).$

𝑓⁡(𝑥)={|𝑥|−1,𝑖⁢𝑓𝑥≠1𝑥3,𝑖⁢𝑓𝑥=1  𝑎=1$f(x)=\{\begin{array}{lll}\left|x\right|-1,\hfill & if\hfill & x\ne 1\hfill \\ x^3,\hfill & if\hfill & x=1\hfill \end{array}a=1$

𝑓⁡(𝑥)={1𝑥+1,𝑖⁢𝑓𝑥=−2(𝑥+1)₂,𝑖⁢𝑓𝑥≠−2  𝑎=−2$f(x)=\{\begin{array}{lll}\frac{1}{x+1},\hfill & if\hfill & x=-2\hfill \\ {(x+1)}^2,\hfill & if\hfill & x\ne -2\hfill \end{array}a=-2$

𝑓⁡(𝑥)={√𝑥+3,𝑖⁢𝑓𝑥<1−3√𝑥,𝑖⁢𝑓𝑥>1  𝑎=1$f(x)=\{\begin{array}{lll}\sqrt{x+3},\hfill & if\hfill & x<1\hfill \\ -\sqrt[3]{x},\hfill & if\hfill & x>1\hfill \end{array}a=1$

lim𝑥→𝑐⁡(𝑓⁡(𝑥)+𝑔⁡(𝑥))$\underset{x\to c}{\lim }\left(f(x)+g(x)\right)$

lim𝑥→𝑐⁡𝑓⁡(𝑥)𝑔⁡(𝑥)$\underset{x\to c}{\lim }\frac{f(x)}{g(x)}$

lim𝑥→𝑐⁡(𝑓⁡(𝑥)⋅𝑔⁡(𝑥))$\underset{x\to c}{\lim }\left(f(x)\cdot g(x)\right)$

lim𝑥→0+⁡𝑓⁡(𝑥),𝑓⁡(𝑥)={3⁢𝑥2+2⁢𝑥+15⁢𝑥+3  𝑥>0𝑥<0$\underset{x\to 0^{+}}{\lim }f(x),f(x)=\{\begin{array}{c}3x^2+2x+1\\ 5x+3\end{array}\begin{array}{c}x>0\\ x<0\end{array}$

lim𝑥→0−⁡𝑓⁡(𝑥),𝑓⁡(𝑥)={3⁢𝑥2+2⁢𝑥+15⁢𝑥+3  𝑥>0𝑥<0$\underset{x\to 0^{-}}{\lim }f(x),f(x)=\{\begin{array}{c}3x^2+2x+1\\ 5x+3\end{array}\begin{array}{c}x>0\\ x<0\end{array}$

lim𝑥→3+⁡(3⁢𝑥−[𝑥])$\underset{x\to 3^{+}}{\lim }\left(3x-[x]\right)$

limℎ→0⁡((ℎ+6)₂−36ℎ)$\underset{h\to 0}{\lim }\left(\frac{{\left(h+6\right)}^2-36}{h}\right)$

lim𝑥→25⁡(𝑥2−625√𝑥−5)$\underset{x\to 25}{\lim }\left(\frac{x^2-625}{\sqrt{x}-5}\right)$

lim𝑥→1⁡(−𝑥2−9⁢𝑥𝑥)$\underset{x\to 1}{\lim }\left(\frac{-x^2-9x}{x}\right)$

lim𝑥→4⁢7−√12⁢𝑥+1𝑥−4$\begin{array}{c}\lim \\ {}^{x\to 4}\end{array}\frac{7-\sqrt{12x+1}}{x-4}$

lim𝑥→−3⁡(13+1𝑥3+𝑥)$\underset{x\to -3}{\lim }\left(\frac{\frac{1}{3}+\frac{1}{x}}{3+x}\right)$

𝑓⁡(𝑥)=−2𝑥−4;𝑎=4$f(x)=\frac{-2}{x-4};a=4$

𝑓⁡(𝑥)=−2(𝑥−4)₂;𝑎=4$f(x)=\frac{-2}{{\left(x-4\right)}^2};a=4$

𝑓⁡(𝑥)=−𝑥𝑥2−𝑥−6;𝑎=3$f(x)=\frac{-x}{x^2-x-6};a=3$

𝑓⁡(𝑥)=6⁢𝑥2+23⁢𝑥+204⁢𝑥2−25;𝑎=−52$f(x)=\frac{6x^2+23x+20}{4x^2-25};a=-\frac{5}{2}$

𝑓⁡(𝑥)=√𝑥−39−𝑥;𝑎=9$f(x)=\frac{\sqrt{x}-3}{9-x};a=9$

𝑓⁡(𝑥)=𝑥2−2⁢𝑥−15$f(x)=x^2-2x-15$

𝑓⁡(𝑥)=𝑥2−2⁢𝑥−15𝑥−5$f(x)=\frac{x^2-2x-15}{x-5}$

𝑓⁡(𝑥)=𝑥2−2⁢𝑥𝑥2−4⁢𝑥+4$f(x)=\frac{x^2-2x}{x^2-4x+4}$

𝑓⁡(𝑥)=𝑥3−1252⁢𝑥2−12⁢𝑥+10$f(x)=\frac{x^3-125}{2x^2-12x+10}$

𝑓⁡(𝑥)=𝑥2−1𝑥2−𝑥$f(x)=\frac{x^2-\frac{1}{x}}{2-x}$

𝑓⁡(𝑥)=𝑥+2𝑥2−3⁢𝑥−10$f(x)=\frac{x+2}{x^2-3x-10}$

𝑓⁡(𝑥)=𝑥+2𝑥3+8$f(x)=\frac{x+2}{x^3+8}$

𝑓⁡(𝑥)=3⁢𝑥+2$f(x)=3x+2$

𝑓⁡(𝑥)=5$f(x)=5$

𝑓⁡(𝑥)=1𝑥+1$f(x)=\frac{1}{x+1}$

𝑓⁡(𝑥)=ln⁡(𝑥)$f(x)=\ln (x)$

𝑓⁡(𝑥)=𝑒2⁢𝑥$f(x)=e^{2x}$

𝑓⁡(𝑥)=4⁢𝑥−6$f(x)=4x-6$

𝑓⁡(𝑥)=5⁢𝑥2−3⁢𝑥$f(x)=5x^2-3x$

Finndu jöfnu snertils við graf 𝑓 ⁡ ( 𝑥 ) f ⁡ ( x ) við tilgreint gildi 𝑥 . x 𝑓 ⁡ ( 𝑥 ) = − 𝑥 3 + 4 ⁢ 𝑥 f ⁡ ( x $f\left(x\right)$) = − x 3 + 4 ⁢ x ; 𝑥 = $x$2 . x =

$f(x)=-x^3+4x$2. $x=2.$

𝑓⁡(𝑥)=𝑥|𝑥|$f(x)=\frac{x}{\left|x\right|}$

Rúmmál réttrar keilu er 𝑉 = 1 3 ⁢ 𝜋 ⁢ 𝑟 2 ⁢ ℎ . $V=\frac{1}{3}\pi r^{2}h$ Gefin keila hefur fasta hæð 9 cm og breytilegan radíus. Finndu augnabliksbreytingarhraða rúmmálsins með tilliti til radíuss þegar radíusinn er 2 cm. Gefðu nákvæmt svar með 𝜋 . $\pi$