## shahzadjalbani 3 years ago {sin7x/sin6x x not = 0 f(x)={6/7 x=0 is function continuous

@physicsme @shadowfiend @anuragtripathi @daniellemmcqueen @LivForMusic @karatechopper

2. physicsme

@ujjwal

@ujjwal

4. physicsme

@supercrazy92

5. ujjwal

Is the question demanding if the function continuous at 0 or not?

Is function is continuous......? @ujjwal

7. 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

9. 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

10. ujjwal

X tends to 0 when x is not equal to 0

11. joemath314159

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:

12. joemath314159

$\lim_{x\rightarrow 0}\frac{\sin(x)}{x}=1$

13. joemath314159

sry, pressed post before i was done.

14. ujjwal

$\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

15. ujjwal

It is already mentioned above but then i would like to post it again. $\lim_{x \to 0}\frac{\sin \theta}{\theta}=1$