Solving Limits Involving Nested Square Roots

In mathematics, limits involving nested square roots can be fascinating and challenging. Let's consider a specific example to illustrate the process of solving such a limit. We aim to solve the limit of the expression ( lim_{x to 0} frac{sqrt{x}}{sqrt{x sqrt{x sqrt{x}}}} ). This problem requires a step-by-step approach to simplify and evaluate the limit.

Simplifying the Expression

First, let's consider the expression inside the limit. We notice that both the numerator and the denominator contain nested square roots. To simplify this, we can start by making a substitution to make the expression more manageable.

Substitution

Let's substitute ( y sqrt{x} ). This substitution will simplify the nested square roots:

First, express the original limit in terms of ( y ):

[ lim_{y to 0} frac{y}{sqrt{y^2 sqrt{y^2 y}}} ]

Next, simplify the denominator:

[ lim_{y to 0} frac{y}{y sqrt{1 frac{sqrt{y^2 y}}{y^2}}} ]

Further simplification gives:

[ lim_{y to 0} frac{1}{sqrt{1 frac{sqrt{y^2 y}}{y^2}}} ]

Since ( y sqrt{x} ), substituting back ( y ) with ( sqrt{x} ) gives:

[ lim_{x to 0} frac{1}{sqrt{1 frac{sqrt{x sqrt{x}}}{x}}} ]

Further simplification under the square root:

[ lim_{x to 0} frac{1}{sqrt{1 sqrt{frac{1}{x} frac{1}{x sqrt{x}}}}} ]

Now, as ( x to 0 ), ( frac{1}{x} ) approaches infinity, making the term under the square root approach infinity. Hence:

[ lim_{x to 0} frac{1}{sqrt{1 sqrt{frac{1}{x} frac{1}{x sqrt{x}}}}} 0 ]

Thus, we have solved the limit:

[ boxed{0} ]

Step-by-Step Insight

Starting from the inside and working our way outward, we simplify the nested square root expression. Recognizing that as ( x to 0 ), ( sqrt{x} ) is approximately ( sqrt{x} ), we can make the expression more familiar and simpler:

Let ( y sqrt{x} ). Then:

( lim_{x to 0} frac{sqrt{x}}{sqrt{x sqrt{x sqrt{x}}}} lim_{y to 0} frac{y}{sqrt{y^2 sqrt{y^2 y}}} )

Simplifying the inner term:

( lim_{y to 0} frac{y}{y sqrt{1 frac{sqrt{y^2 y}}{y^2}}} )

Further simplification:

( lim_{y to 0} frac{1}{sqrt{1 frac{sqrt{y^2 y}}{y^2}}} )

Recognizing the behavior of the terms as ( y to 0 ), the limit evaluates to 0.

Conclusion

By following these detailed steps, we can see that the limit of the given expression is 0. This process illustrates the importance of simplification and substitution in solving complex mathematical expressions.

Key takeaways:

Substitution can simplify expressions. Recognizing the behavior of terms as variables approach certain values is crucial. Breaking down nested functions step-by-step can make them more understandable.

These techniques are fundamental in solving limits involving nested square roots or other complex mathematical expressions.