设函数f(x)=(1-cosx)/x^2(x>0);=k(x=0);=(1/x-1/(e^x-1))(x

Question

设函数f(x)=(1-cosx)/x^2(x>0);=k(x=0);=(1/x-1/(e^x-1))(x<0)当k为何值时,在x=0处连续.

Answer 1

f(x)在x=0处函数值为k,由连续性知,在x=0处极限值等于函数值,以求右极限为例,f(x)=(1-cosx)/x^2(x>0);当x趋近于0时,此式子极限为1/2(洛必达法则或者等价无穷小）.同理求的左极限应该也为1/2,所以k=1/2