How do you find the limit #(x^-1+x^(-1/2))/(x+x^(-1/2))# as #x->1^+#?

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Answer 1 · 2016-11-01T20:43:04

Always try substitution first.

#1^r=1# for any real number #r#.

So

#lim_(xrarr1^+)(x^-1+x^(-1/2)) = 1+1=2#

Similarly,

#lim_(xrarr1^+)(x+x^(-1/2)) = 1+1=2#.

Therefore,

#lim_(xrarr1^+)(x^-1+x^(-1/2)) /(x+x^(-1/2)) = 2/2 = 1#