L I M I E T E N
met logaritmen en getal e

LIMITS (involving log or Euler's number (e)
13 \(\left ( 1+\frac 1x \right )^{x+3} \) 10 \(\frac {n\,+\,1} {n} \) en \( \left(\frac {n\,+\,1} {n}\right)^n \)
12 \(\sqrt[x]{x+1}\) 14 \(\,\log_3\,(x+1)^2 \)
11 \(\log \,(2x) \) 46 \(\sqrt[x]{x}\)
20 \(\left ( \frac{x\,+\,6}{x} \right )^{3x} \) 21 \(\left ( \frac{1\,+\,x}{x} \right )^{4-3x}\)
24 \(\left ( 1-\frac 3x \right )^{-6x} \) 22 \(\left ( 1-\frac 2x \right )^{4x}\)
25 \(\left ( \frac{x\,-\,3}{x} \right )^{\frac x2} \) 23 \(\left (1- \frac{1}{2x} \right )^{-3x}\)
26 \(\;\left ( 1\!-ax \right )^{\frac 1x} \) 28 \( (1+x \sqrt2)^{\frac {\sqrt 2} {x}} \)
32 \(\sqrt[x-1]{x}\) 34 lim   \( (1+\cos^2 x)^{1+\tan^2 x}\)
27 \(\large \frac {4^x\:-\:\left(\frac 12\right)^x} {x} \) 30 \(\large \left( \frac {\ln x} {x}+ \frac{x}{e^x} \right)\)
29 \(\large \frac {x^x} {(x\,-\,1)^x} \) 37 \(\large \frac {1\,-\:\cos x} {1\,-\,x\,-\,e^{-x}} \)
43 \(\log_x 2 \) 44 \(\log_{\frac 12} x\)
31 \(\large x^{-\frac 2x} \) 33 \(\large \frac {e^2.\,x^2} {e^x} \)
39 \(\large \frac {e^x\,-\,1} {ex\,-\,1} \) 38 \(\large \frac {2^{x+2}-16} {4^x\,-\,16} \)
42 \(\ln\,(\tan x)\) 35 \(\;(\tan 2x.\ln \tan x)\)
46 \(\large \frac {\log x} {x\,-\,1} \) 47 \(\large \frac {\log_4 x} {\log_8 x} \)
41 \(\large \frac {\sqrt x} {\ln x} \) 40 \(\large \frac {e^{2+h}-\,e^2} {h} \)
45 \(\log_{0,3} \,e^x \) 46 \(\large \frac {\log_2\, (1\,-\,x^2)} {\log_4\, (\cos x)} \)

48 \( \left ( \frac{\sin {2x}}{ \sin x}+\frac{\log\; \sin {2x}}{\log\; \sin x} \right )\) new
49  (sin 1)x (tan 1)x e−x
50\(\large \frac {\ln(1\,+\,2x^2\,+\,4x^4)} {\ln(1\,+\,3x^4\,+\,9x^8)} \)
60 \(\large \frac{x^x}{(x\,+\,1)^x} \)
61 \(\large (\ln x)^{\frac {1} {x-e}} \)
62 \((\cos x)^{\frac {1} {\sin^{{\small 2}} x}} \)
63 \((\sin x + \cos x)^{\frac {1} {2x}} \)
64 \(\large \left( \frac {\tan x} {x} \right)^{\frac {1}{x^{\small 2}}} \)
65 \(\large \frac {(1\,+\,x)^{\frac 1x}\,-\,e} {x} \)
66 \(\left( e^{\frac 1x} + \frac 1x \right)^x \)
67 \(\large \left( \frac{\sin x}{x} \right) ^{\frac {1} {x^{\small 2}}} \)
68 \(\large \frac {\ln\,(1\,-\,x^2)} {\ln \cos x} \)
69 \(\large \left( \frac {x\,-\,2} {x\,-\,1} \right)^{x-1} \)
70 \(\large \left(\frac {3^x\,+\:4^x} {2}\right)^\frac 1x \)
71 \(\large \left(\frac {1\,+\,2x} {1\,+\,x}\right)^\frac{x-1}{x} \)
72 \( \left(1+2+3\,+\,...+\,n\right )^\frac 1n \)
73 \(\large \left ( x^2-x \right )^{\frac 2x}\)
74 \(\large \frac{e^x\:-\;x\;-\:1}{x^2}\) new


afgeleiden (enkel berekenen)

derivatives (calculation)

extremumvraagstukken die zonder
afgeleiden kunnen worden opgelost

extremum problems (without derivatives)

extremumvraagstukken die met
afgeleiden 'moeten' worden opgelost

extremum problems (with derivatives)

toepassingen afgeleiden (b.v. raaklijn)

applications derivatives (e.g. tangent line)

max./min. bij goniometrische uitdrukkingen

max/min for trigonometric expressions

limieten van veeltermbreuken (a)

limits of polynomial fractions (a)

limieten van veeltermbreuken (b)

limits of polynomial fractions (b)

limieten met één wortelvorm

limits involving one square root

limieten met meerdere wortelvormen

limits involving more than one square root

limieten van goniometrische functies

limits involving trigonometric functions

limieten met logaritme of het getal e

limits of exponential or logarithmic functions

verticale en horizontale asymptoten

vertical and horizontal asymptotes

schuine asymptoten - VARIA

oblique asymptotes - VARIA

telling vanaf ma 2 sept 2013