Question

lim X tends to infinity (x^1/2 - xln(1+1/x^1/2)​

Answer 1

Answer:

lim

x

→

∞

x

−

ln

(

x

2

+

1

)

=

+

∞

Explanation:

As

x

=

ln

e

x

, we can write the function as:

x

−

ln

(

x

2

+

1

)

=

ln

e

x

−

ln

(

x

2

+

1

)

Using the properties of logarithms then we have:

x

−

ln

(

x

2

+

1

)

=

ln

(

e

x

x

2

+

1

)

Now we know the the exponential is an infinite of higher order than any polynomial for

x

→

∞

, so

lim

x

→

∞

e

x

x

2

+

1

=

+

∞

and then as the logarithm is continuous in

(

0

,

+

∞

)

lim

x

→

∞

ln

(

e

x

x

2

+

1

)

=

+

∞

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