Given e>0 find an interval I=(5,5+delta ),delta gt 0 such that if x lies in I, then sqrt [4](x-5)lt e What limit is being verified and what is its value? If x lies in the interval I=(5,5+8) then sqrt [4](x-5)lt e when 0lt delta lt square (Type an expression using eas the variable )

## Step 1The problem is asking us to find an interval \(I=(5,5+\delta)\), where , such that if lies in , then . This is a limit problem, and we are verifying the limit of the function \(f(x) = \sqrt[4]{x-5}\) as approaches 5.## Step 2The limit we are verifying is . We need to find a value for such that for all in the interval \(I=(5,5+\delta)\), the inequality holds.## Step 3To find the value of , we need to solve the inequality . This can be done by raising both sides of the inequality to the power of 4, which gives us \(x-5 < e^).## Step 4Solving for , we get . This means that the interval must be less than .## Step 5Therefore, the value of must be less than , which simplifies to .