Proceeding directly from the definition ,ascertain that the straight line `y = 2x + 1 ` is an asymptote of the curve ` y = (2x^(4)+x^(3)+1)/(x^(3))`

Answer

To ascertain that the straight line \( y = 2x + 1 \) is an asymptote of the curve \( y = \frac{2x^4 + x^3 + 1}{x^3} \), we will proceed directly from the definition of an asymptote.

### Step 1: Identify the function and the asymptote
We have the function:
\[ f(x) = \frac{2x^4 + x^3 + 1}{x^3} \]
and the proposed asymptote:
\[ y = 2x + 1 \]

### Step 2: Set up the limit to find the asymptote
To check if \( y = 2x + 1 \) is an asymptote, we need to evaluate the limit:
\[ \lim_{x \to \infty} \left( f(x) - (2x + 1) \right) \]

### Step 3: Substitute \( f(x) \) into the limit expression
Substituting \( f(x) \) into the limit gives:
\[ \lim_{x \to \infty} \left( \frac{2x^4 + x^3 + 1}{x^3} - (2x + 1) \right) \]

### Step 4: Simplify the expression
We can rewrite the limit as:
\[ \lim_{x \to \infty} \left( \frac{2x^4 + x^3 + 1 - (2x^4 + x^3)}{x^3} \right) \]
This simplifies to:
\[ \lim_{x \to \infty} \left( \frac{1}{x^3} \right) \]

### Step 5: Evaluate the limit
Now, we evaluate the limit:
\[ \lim_{x \to \infty} \frac{1}{x^3} = 0 \]

### Step 6: Conclusion
Since the limit is finite (specifically, it equals 0), we conclude that the straight line \( y = 2x + 1 \) is indeed an asymptote of the curve \( y = \frac{2x^4 + x^3 + 1}{x^3} \).

### Final Answer
Thus, we have shown that the line \( y = 2x + 1 \) is an asymptote of the curve.

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