Kalkulus – Contoh Soal dan Jawaban Limit

Contoh 1

\[ \lim_{x\rightarrow \frac{\pi}{4}}\frac{cos x-sin x}{cos 2x} \] \[ =\lim_{x\rightarrow \frac{\pi}{4}}\frac{cos x-sin x}{cos^{2}x-sin^{2}x} \] \[ \lim_{x\rightarrow \frac{\pi}{4}}\frac{cos x-sin x}{(cos x – sin x)(cos x+ sin x)} \] \[ \lim_{x\rightarrow \frac{\pi}{4}}\frac{1}{(cos x + sin x)} \] \[ =\frac{1}{2}\sqrt{2} \]

Contoh 2

\[ \lim_{x\rightarrow 0}\frac{3.sin4x}{5.tan6x} \] \[ =\frac{3.4x}{5.6x} \] \[ =\frac{2}{5} \]

Contoh 3

\[ \lim_{x\rightarrow 0}\frac{1-cos4x}{x.tan3x} \] \[ =\lim_{x\rightarrow 0}\frac{1-(1-2sin^{2}2x}{x.tan3x} \] \[ =\lim_{x\rightarrow 0}\frac{2sin^{2}2x}{x.tan3x} \] \[ \frac{2(2x)^{2}}{x.3x} \] \[ =\frac{8}{3} \]

Contoh 4

\[ lim_{x\rightarrow a}\frac{(2a+x)sin(a-x)}{x^{2}-a^{2}} \] \[ =lim_{x\rightarrow a}\frac{(2a+x)sin(a-x)}{-(x+a)(x+a)} \] \[ =\frac{(2a+a)}{-(a+a)} \] \[ =-\frac{3}{2} \]

Contoh 5

\[ \lim_{x\rightarrow 3}\frac{x^{4}-9x^{2}}{sin(9-x^{2})} \] \[ =\lim_{x\rightarrow 3}\frac{-x^{2}(-x^{2}+9)}{sin(9-x^{2})} \] \[ =-(3)^{2} \] \[ =-9 \]

Contoh 6

\[ \lim_{x\rightarrow \frac{\pi}{2}}\frac{1+cos2x}{cosx} \] \[ =\lim_{x\rightarrow \frac{\pi}{2}}\frac{1+2cos^{2}x-1}{cosx} \] \[ =\lim_{x\rightarrow \frac{\pi}{2}}2cosx \] \[ =0 \]

Materi Lengkap

Berikut adalah beberapa materi lengkap yang membahas tuntas mengenai Limit.


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