To find the limit of the given function as $x$ approaches infinity, let's analyze the expression step by step:

The expression is:

$\lim_{x \to \infty} \left(49 - \frac{x}{7 + x}\right)$

Step 1: Simplify the fraction $\frac{x}{7 + x}$

Let's rewrite the fraction $\frac{x}{7 + x}$:

$\frac{x}{7 + x} = \frac{x}{x \left(\frac{7}{x} + 1\right)} = \frac{1}{\frac{7}{x} + 1}$

As $x$ approaches infinity, $\frac{7}{x}$ approaches 0. So, the fraction simplifies to:

$\frac{1}{\frac{7}{x} + 1} \approx \frac{1}{1} = 1$

Thus, the expression inside the limit becomes:

$\lim_{x \to \infty} \left(49 - 1\right)$

Step 2: Final computation

Simplifying the expression:

$\lim_{x \to \infty} (49 - 1) = 48$

Final Answer:

$\lim_{x \to \infty} \left(49 - \frac{x}{7 + x}\right) = 48$

Would you like more details, or do you have any further questions?

Here are some related questions:

1. What happens to $\lim_{x \to \infty} \frac{7x}{7+x}$?
2. How would you compute $\lim_{x \to 0} \frac{x^2}{x+7}$?
3. How do you evaluate the limit of a sum of functions as $x \to \infty$?
4. What is the difference between infinite limits and limits at infinity?
5. How would you find the limit of a product of functions as $x$ approaches infinity?

Tip: When dealing with limits as $x$ approaches infinity, it's often helpful to divide by the highest power of $x$ in the denominator to simplify the expression.