find the limit as x approaches infinity of sqrt(49x^2+x)-7x

To solve the limit of \( \lim_{x \to \infty} \left( \sqrt{49x^2 + x} - 7x \right) \), let's break it down step by step.

### Step 1: Factor the expression inside the square root
We first factor out \(x^2\) from the square root:

\[
\sqrt{49x^2 + x} = \sqrt{x^2(49 + \frac{1}{x})} = x\sqrt{49 + \frac{1}{x}}
\]

Now, rewrite the expression:

\[
\lim_{x \to \infty} \left( \sqrt{49x^2 + x} - 7x \right) = \lim_{x \to \infty} \left( x\sqrt{49 + \frac{1}{x}} - 7x \right)
\]

### Step 2: Simplify the expression
Factor out \(x\) from the expression:

\[
\lim_{x \to \infty} x \left( \sqrt{49 + \frac{1}{x}} - 7 \right)
\]

### Step 3: Evaluate the limit
As \(x \to \infty\), the term \(\frac{1}{x}\) approaches zero. So, inside the square root:

\[
\sqrt{49 + \frac{1}{x}} \to \sqrt{49} = 7
\]

Thus, the expression inside the parentheses becomes:

\[
\sqrt{49 + \frac{1}{x}} - 7 \to 7 - 7 = 0
\]

### Step 4: Apply the limit
The whole expression tends to:

\[
\lim_{x \to \infty} x \cdot 0 = 0
\]

### Final Answer:
\[
\lim_{x \to \infty} \left( \sqrt{49x^2 + x} - 7x \right) = 0
\]

Would you like more details on any part of the process?

Here are 5 related questions to consider:
1. How would the limit change if the coefficient of \(x^2\) was different, say \(36x^2\)?
2. How do you handle limits involving square roots and infinity in general?
3. What happens if the term inside the square root includes higher powers of \(x\)?
4. Can you use L'Hopital's Rule to verify this limit?
5. How do limits at infinity differ from limits at finite values?

**Tip:** When dealing with limits involving square roots at infinity, factoring out the highest power of \(x\) inside the square root is a helpful technique!

Find the limit as x approaches infinity of sqrt(49x^2 + x) - 7x

Explore the step-by-step solution for finding the limit of sqrt(49x^2 + x) - 7x as x approaches infinity. Learn how to simplify square roots, apply limit laws, and handle infinity calculations. Suitable for advanced high school or college-level calculus students.

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