Limits_1 You should use theorems on limits and the result that \[ \lim_{{@v@}\rightarrow 0}{\left(\frac{\sin({@v@})}{@v@}\right)}=1. \]
You should not use a Calculator or L'Hopital's rule.

Evaluate the following limit: \[ \lim_{{@v@}\rightarrow 0}{\left(\frac{\sin({@n1@} {@v@})}{\sin({@n2@}{@v@})}\right)} .\]
The limit = [[input:ans1]] [[validation:ans1]]

]]> We have \[ \lim_{{@v@}\rightarrow 0}{\left(\frac{\sin({@n1@} {@v@})}{\sin({@n2@}{@v@})}\right)} = \lim_{{@v@}\rightarrow 0}{\left(\frac{\sin({@n1@} {@v@})}{{@n1@} {@v@}} \frac{{@n2@} {@v@}}{\sin({@n2@}{@v@})} \frac{{@n1@}{@v@}}{{@n2@}{@v@}}\right)} = 1\cdot1\cdot\frac{@n1@}{@n2@}= {@correctanswer@}\]

]]> 1 0 0 v:x; n1:rand_with_step(2,9,1); n2:rand_with_step(11,19,1); f:sin(n1*v)/sin(n2*v); correctanswer:n1/n2; [[feedback:Result]]

]]> \(\lim_{{@v@}\rightarrow 0}{\left(\frac{\sin({@n1@} {@v@})}{\sin({@n2@}{@v@})}\right)}\) = {@correctanswer@} 1 0 0 *10 dot 1 i cos-1 lang [ 0 {"author":[{"firstName":"","lastName":"","institution":"Unknown - Legacy question","year":"2025"}],"language":["en"],"isPartOf":"STACK Library","license":"cc-sa-4.0","additional":{"UoE":{"Notes":"This is automatically generated metadata."}},"freeform":{}} ans1 algebraic correctanswer 20 1 0 0 limit 1 1 1 1 1 Result 1.0000000 1 1 0 AlgEquiv ans1 correctanswer 0 = 1.0000000 0 -1 Result-0-T = 0.0000000 0 -1 Result-0-F 755191200 1987251928 361610374 548169882 1504426525 1753941928 1021896733 1000836762 1893832123 162057896 812124702 2077406899 1666608699 1688793199 2011396505 1706274855 1 ans1 correctanswer Result 1.0000000 0.0000000 Result-0-T 2 ans1 n1 Result 0.0000000 0.0000000 Result-0-F 3 ans1 n2 Result 0.0000000 0.0000000 Result-0-F