How do you find the limit of the cubic root of x^2+1 all mines x ? Thank you.

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Answer 1 · 2016-11-15T15:19:57

The question does not say what number #x# is approaching.

The properties of limits assure us that for any number #a#,

#lim_(xrarra)(root(3)(x^2+1)-x)# can be found by substituting #a# in place of #x# and doing the arithmetic.

For any #a#, the subtraction property of limits gets us:

#lim_(xrarra)(root(3)(x^2+1)-x) = lim_(xrarra)root(3)(x^2+1)-lim_(xrarra)x #

The power (or root) property of limits lets us continue:

# = root(3)(lim_(xrarra)(x^2+1))-lim_(xrarra)x #

Using the addition property:

# = root(3)(lim_(xrarra)(x^2)+lim_(xrarra)1)-lim_(xrarra)x #

And the power property again

# = root(3)((lim_(xrarra)x)^2)+lim_(xrarra)1)-lim_(xrarra)x #

Now use the fact that #lim_(xrarra)x = a# to write

# = root(3)(a^2+1)-a#