How do you find the limit #lim (root4(x+1)-root4x)x^(3/4)# as #x-oo#?

Question

How do you find the limit #lim (root4(x+1)-root4x)x^(3/4)# as #x->oo#?

Ezra Adams · Accepted Answer

#1/4#

#lim_(x to oo) (root4(x+1)-root4x)x^(3/4)#

#= lim_(x to oo) x^(1/4)(root4(1+1/x))-1)x^(3/4)#

#= lim_(x to oo) x ( root4(1+1/x)-1 )#

By Biniomial Expansion:

#= lim_(x to oo) x((1+1/(4x) + O(1/x)^2)-1 )#

#= lim_(x to oo) 1/(4) + O(1/x)^1 = 1/4#

Isaiah Allen · Answer

#1/4#

#root(4)(x+1)-root(4)(x)=(root(4)((x+1)^2)-root(4)(x^2))/(root(4)(x+1)+root(4)(x)) = (root(4)((x+1)^4)-root(4)(x^4))/((root(4)(x+1)+root(4)(x))(root(4)((x+1)^2)+root(4)(x^2)))=1/((root(4)(x+1)+root(4)(x))(root(4)((x+1)^2)+root(4)(x^2)))=1/(x^(3/4)(root(4)(1+1/x)+root(4)(1))(root(4)((1+1/x)^2)+root(4)(1)))#

then

#lim_(x->oo)(root(4)(x+1)-root(4)(x))x^(3/4) = 1/4#

Paisley Johnson · Answer

undefined To find the limit of the expression lim (root4(x+1)-root4x)x^(3/4) as x approaches infinity, we can simplify the expression first.

Using the property of radicals, we can simplify root4(x+1) - root4x as (x+1)^(1/4) - x^(1/4).

Next, we multiply this simplified expression by x^(3/4).

Now, we have the expression (x+1)^(1/4) - x^(1/4) multiplied by x^(3/4).

To evaluate the limit as x approaches infinity, we can use the fact that the dominant term in the expression will determine the limit.

Since x^(3/4) grows faster than (x+1)^(1/4) and x^(1/4), the dominant term is x^(3/4).

Therefore, as x approaches infinity, the limit of the expression is infinity.

How do you find the limit #lim (root4(x+1)-root4x)x^(3/4)# as #x-&gt;oo#?

How do you find the limit #lim (root4(x+1)-root4x)x^(3/4)# as #x->oo#?