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

Answer 1

Ezekiel Alexander

Always try substitution first.

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

#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#

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Answer 2

Ezra Adams

To find the limit of (x^-1+x^(-1/2))/(x+x^(-1/2)) as x approaches 1 from the positive side, we can simplify the expression by multiplying the numerator and denominator by x^(1/2). This gives us (x^(1/2)+1)/(x^(3/2)+x).

Now, substituting x=1 into the simplified expression, we get (1^(1/2)+1)/(1^(3/2)+1). Simplifying further, we have (1+1)/(1+1), which equals 2/2.

Therefore, the limit of (x^-1+x^(-1/2))/(x+x^(-1/2)) as x approaches 1 from the positive side is 1.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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

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