The value of `lim_(x->oo)(((x-1)(x-2)(x+3)(x+10)(x+15))^(1/5)-x)` is ....

Answer

To solve the limit \( \lim_{x \to \infty} \left( ((x-1)(x-2)(x+3)(x+10)(x+15))^{1/5} - x \right) \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the limit:
\[ \lim_{x \to \infty} \left( ((x-1)(x-2)(x+3)(x+10)(x+15))^{1/5} - x \right) \]

### Step 2: Factor out \( x \) from each term inside the product
We can factor \( x \) out of each term in the product:
\[ (x-1) = x(1 - \frac{1}{x}, \quad (x-2) = x(1 - \frac{2}{x}), \quad (x+3) = x(1 + \frac{3}{x}), \quad (x+10) = x(1 + \frac{10}{x}), \quad (x+15) = x(1 + \frac{15}{x}) \]
Thus, we have:
\[ (x-1)(x-2)(x+3)(x+10)(x+15) = x^5 \left( (1 - \frac{1}{x})(1 - \frac{2}{x})(1 + \frac{3}{x})(1 + \frac{10}{x})(1 + \frac{15}{x}) \right) \]

### Step 3: Substitute back into the limit
Now substituting back into the limit gives:
\[ \lim_{x \to \infty} \left( \left( x^5 \left( (1 - \frac{1}{x})(1 - \frac{2}{x})(1 + \frac{3}{x})(1 + \frac{10}{x})(1 + \frac{15}{x}) \right) \right)^{1/5} - x \right) \]

### Step 4: Simplify the expression
Taking the fifth root of \( x^5 \) gives \( x \):
\[ \lim_{x \to \infty} \left( x \left( (1 - \frac{1}{x})(1 - \frac{2}{x})(1 + \frac{3}{x})(1 + \frac{10}{x})(1 + \frac{15}{x}) \right)^{1/5} - x \right) \]
This simplifies to:
\[ \lim_{x \to \infty} x \left( (1 - \frac{1}{x})(1 - \frac{2}{x})(1 + \frac{3}{x})(1 + \frac{10}{x})(1 + \frac{15}{x})^{1/5} - 1 \right) \]

### Step 5: Expand the product using the approximation
Using the approximation \( (1 + u)^{n} \approx 1 + nu \) for small \( u \):
\[ (1 - \frac{1}{x})(1 - \frac{2}{x})(1 + \frac{3}{x})(1 + \frac{10}{x})(1 + \frac{15}{x}) \approx 1 - \frac{1}{x} - \frac{2}{x} + \frac{3}{x} + \frac{10}{x} + \frac{15}{x} \]
Combining these gives:
\[ 1 + \left( -\frac{1}{x} - \frac{2}{x} + \frac{3}{x} + \frac{10}{x} + \frac{15}{x} \right) = 1 + \frac{25}{x} \]

### Step 6: Substitute back into the limit
Now substituting back gives:
\[ \lim_{x \to \infty} x \left( \left(1 + \frac{25}{x}\right)^{1/5} - 1 \right) \]
Using the approximation again:
\[ \left(1 + \frac{25}{x}\right)^{1/5} \approx 1 + \frac{5}{5} \cdot \frac{25}{x} = 1 + \frac{5}{x} \]
Thus, we have:
\[ \lim_{x \to \infty} x \left( \frac{5}{x} \right) = 5 \]

### Final Answer
Therefore, the value of the limit is:
\[ \boxed{5} \]