Here's the question you clicked on:
shahzadjalbani
{sin7x/sin6x x not = 0 f(x)={6/7 x=0 is function continuous
@physicsme @shadowfiend @anuragtripathi @daniellemmcqueen @LivForMusic @karatechopper
Is the question demanding if the function continuous at 0 or not?
Is function is continuous......? @ujjwal
do you know they define continuity at a point? Here the question should be, "is the function continuous at x=0?"
yes this is the question.... @ujjwal
To test the continuity of function at x=0, find the limit when x--->0 And the find f(0). If \[\\lim_{x \to 0}f(x)=f(0)\] the the function is continuous at x=0
X tends to 0 when x is not equal to 0
Lets re-write the function:\[\frac{\sin(7x)}{\sin(6x)}=\frac{\frac{\sin(7x)}{7x}}{\frac{\sin(6x)}{7x}}\]Now multiply the numerator and denominator by 7/6:\[\frac{\frac{7}{6}\cdot \frac{\sin(7x)}{7x}}{\frac{7}{6}\cdot \frac{\sin(6x)}{7x}}=\frac{\frac{7}{6}\cdot \frac{\sin(7x)}{7x}}{\frac{\sin(6x)}{6x}}\]Now you can take the limit as x goes to 0 using the fact that:
\[\lim_{x\rightarrow 0}\frac{\sin(x)}{x}=1\]
sry, pressed post before i was done.
\[\lim_{x \to 0}f(x)=\lim_{x \to 0}\frac{\sin 7x}{\sin 6x}\]\[=\lim_{x \to 0}\frac{\frac{\sin 7x}{7x}}{\frac{\sin 6x}{6x}}\times \frac{6}{7}\]\[=\frac{6}{7}\] Also, f(0)=6/7 since\[\lim_{x \to 0}f(x)=f(0)\]The function is continuous at x=0
It is already mentioned above but then i would like to post it again. \[\lim_{x \to 0}\frac{\sin \theta}{\theta}=1\]