Trigonometrijski limes kad x teži jednoj polovini

TEKST ZADATKA

Odrediti graničnu vrednost:

\lim_{{x} \to {\frac12}}\bigg(\frac{\sin(2x-1)}{4x^2-1}+\frac{8x^3-1}{2x^2-x}\bigg)

REŠENJE ZADATKA

Primeniti formulu za razliku kubova: $a^3-b^3=(a-b)(a^2+ab+b^2)$

\lim_{{x} \to {\frac12}}\bigg(\frac{\sin(2x-1)}{4x^2-1}+\frac{(2x-1)(4x^2+2x+1)}{x(2x-1)}\bigg)

Primeniti formulu za razliku kvadrata: $a^2-b^2=(a-b)(a+b)$

\lim_{{x} \to {\frac12}}\bigg(\frac{\sin(2x-1)}{(2x-1)(2x+1)}+\frac{(2x-1)(4x^2+2x+1)}{x(2x-1)}\bigg) \\ \lim_{{x} \to {\frac12}}\bigg(\frac{\sin(2x-1)}{(2x-1)(2x+1)}+\frac{4x^2+2x+1}{x}\bigg)

Raščlaniti izraz.

\lim_{{x} \to {\frac12}}\frac{\sin(2x-1)}{2x-1}\cdot\lim_{{x} \to {\frac12}}\frac1{2x+1}+\lim_{{x} \to {\frac12}}\frac{4x^2+2x+1}{x}

Primeniti tablični limes: $\lim_{{x} \to {0}}\frac {\sin{x}} {x}=1$

1\cdot\lim_{{x} \to {\frac12}}\frac1{2x+1}+\lim_{{x} \to {\frac12}}\frac{4x^2+2x+1}{x}

Zameniti $x=\frac12.$

\frac1{2\cdot\frac12+1}+\frac{4\cdot(\frac12)^2+2\cdot\frac12+1}{\frac12} \\ \frac{13}2

$\lim_{{x} \to {\frac12}}\big(\frac{\sin(2x-1)}{4x^2-1}+$