设f （x）在（-∞，+∞）内有定义，且limx→∞f(x)＝a，g（x）=f(1x)， x≠00， x＝0，则... f(x)在(-∞,+∞)内连续,且limx→∞f（x）存在,...

\u8bbef(x),g(x)\u5728(a,\u6b63\u65e0\u7a77\u5927)\u4e0a\u6709\u5b9a\u4e49,\u4e14lim(x\u8d8b\u4e8e\u6b63\u65e0\u7a77\u5927)f(x)=\u6b63\u65e0\u7a77\u5927\uff0cli

\u6765\u7167\u7247

\u8bbelimf\ufe59x\ufe5a\uff1dA\u3000\u3000\u3000\u3000\u3000\u3000\u3000\ufe59x\u8d8b\u4e8e\u65e0\u7a77\u5927\ufe5a\u2234\u4efb\u610f\u03b5\u3000\u3000\u3000\u3000\u3000\u3000\u5b58\u5728X\uff1eA\u3000\u3000\u3000\u3000\u3000\u3000\u3000\u3000\u3000\u5f53x\uff1eX\u65f6\u3000\u3000\u3000\u3000\u3000\u3000\u3000\uff5cf\ufe59x\ufe5a\uff0dA\uff5c\uff1c\u03b5\uff06\uff0347\uff1b4\u3000\u3000\u3000\u3000\u3000\u3000\u3000\u3000\u3000\u3000\u3000\u3000\u2234\u5bf9\u4efb\u610fx\uff06\uff038321\uff1b\u3001x\uff06\uff038322\uff1b\u2208\ufe59X519\ufe62\u221e\ufe5a\u3000\u6709\uff5cf\ufe59x\uff06\uff038321\uff1b\ufe5a\uff0df\ufe59x\uff06\uff038322\uff1b\ufe5a\uff5c\u2264\uff5cf\ufe59x\uff06\uff038321\uff1b\ufe5a\uff0dA\uff5c\uff0b\uff5cf\ufe59x\uff06\uff038322\uff1b\ufe5a\uff0dA\uff5c\uff1c\u03b5\uff06\uff0347\uff1b2\u3000\u3000\u3000\u3000\u7531\u5eb7\u6258\u5b9a\u7406\u3000\u3000\u3000\u3000\u3000\u3000\u3000\u3000\u3000\u3000f\ufe59x\ufe5a\u5728\uff3ba17X\uff3d\u4e00\u81f4\u8fde\u7eed\u3000\u3000\u3000\u3000\u56e0\u800c\u5b58\u5728\u03b4\uff1cX\uff0da\u3000\u3000\u3000\u3000\u3000\u3000\u3000\u4f7f\uff5cx\uff06\uff038321\uff1b\uff0dx\uff06\uff038322\uff1b\uff5c\uff1c\u03b4\uff0cx\uff06\uff038321\uff1b\uff0cx\uff06\uff038322\uff1b\u2208\uff3ba\uff0cX\uff3d\u65f6\u3000\u3000\u3000\uff5cf\ufe59x\uff06\uff038321\uff1b\ufe5a\uff0df\ufe59x\uff06\uff038322\uff1b\ufe5a\uff5c\uff1c\u03b5\uff06\uff0347\uff1b2\u3000\u3000\u3000\u3000\u3000\u3000\u3000\u4ece\u800c\u5bf9\u4efb\u610fx\uff06\uff038321\uff1b3173x\uff06\uff038322\uff1b\u2208\uff3ba\ufe62\u221e\ufe5a\u53ea\u8981\uff5cx\uff06\uff038321\uff1b\uff0dx\uff06\uff038322\uff1b\uff5c\uff1c\u03b4\u3000\u3000\u3000\u3000\u3000\u3000\u3000\u3000\u5c31\u6709\uff5cf\ufe59x\uff06\uff038321\uff1b\ufe5a\uff0df\ufe59x\uff06\uff038322\uff1b\ufe5a\uff5c\uff1c\u03b5\uff06\uff0347\uff1b2\uff0b\u03b5\uff06\uff0347\uff1b2\uff1d\u03b5\u3000\u3000\u3000\u3000\u3000\u3000\u3000\u3000\u3000\u3000\u3000\u3000\u3000\u3000\u3000\u3000\u2234\u5176\u4e00\u81f4\u8fde\u7eed

因为

lim

x→0

g(x)＝

lim

x→0

f(

1

x

)＝

lim

u→∞

f(u)=a（令u＝

1

x

），又g（0）=0，所以，
①当a=0时，

lim

x→0

g(x)＝g(0)，即g（x）在点x=0处连续；
②当a≠0时，

lim

x→0

g(x)≠g(0)，即x=0是g（x）的第一类间断点．
因此，g（x）在点x=0处的连续性与a的取值有关．
故选：D．

选D，详情如图所示

扩展阅读：∞软件 ... 求解lim x ∞ x n ... ∞x∞ ... limxsin1 x的极限x ∞ ... ∞图片 ... lim x ∞ ... ∞和∞1哪个更大 ... x趋于+∞ ... x∞model ...