这是要看x的取值范围的。分为以下两种情况。
1：当x无限趋近于0是，sinx/x=1
这是高等数学书上的定理。
2：而当x无限趋近于无穷的时候，sinx/x=0.
这个时候可以把x当做无穷小的一个数，而sinx是有界函数，其范围为【-1--1】。
图一为正弦函数，图二为余弦函数。
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无穷小的函数*有界函数，结果自然是无穷小。
即结果为0.
扩展阅读：
函数的传统定义
一般的，在一个变化过程中，假设有两个变量x、y，如果对于任意一个x都有唯一确定的一个y和它对应，那么就称x是自变量，y是x的函数。x的取值范围叫做这个函数的定义域，相应y的取值范围叫做函数的值域。
函数的近代定义
设A，B是非空的数集，如果按照某种确定的对应关系f，使对于集合A中的任意一个数x，在集合B中都有唯一确定的数
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和它对应，那么就称映射
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为从集合A到集合B的一个函数，记作
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或
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其中x叫作自变量，
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叫做x的函数，集合
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叫做函数的定义域，与x对应的y叫做函数值，函数值的集合
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叫做函数的值域，
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叫做对应法则。其中，定义域、值域和对应法则被称为函数三要素
定义域，值域，对应法则称为函数的三要素。一般书写为
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若省略定义域，一般是指使函数有意义的集合。
参考资料：
/baike.baidu.com/item/函数/301912"target="_blank">百度百科--函数