The expression you've provided can be written as:

\[
\lim_{x \to 0} \frac{x + 64 - 8}{x}
\]

Let's simplify the expression inside the limit step-by-step.

\[
\frac{x + 64 - 8}{x} = \frac{x + 56}{x}
\]

Now, split the terms in the fraction:

\[
\frac{x + 56}{x} = \frac{x}{x} + \frac{56}{x} = 1 + \frac{56}{x}
\]

Now, take the limit as \( x \to 0 \):

\[
\lim_{x \to 0} \left( 1 + \frac{56}{x} \right)
\]

The term \( \frac{56}{x} \) tends to infinity as \( x \to 0 \), meaning the whole expression tends to infinity.

Thus, the limit does not exist (DNE) because it tends to infinity.

Would you like more details on how I approached this, or do you have any questions?

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Here are 5 related questions for further exploration:
1. What is the limit when the numerator is changed to \(x + 64 - 8x\)?
2. How do we handle limits that tend toward infinity?
3. What is the difference between limits approaching 0 from the left vs the right?
4. Can L'Hopital's Rule be applied to find limits approaching infinity?
5. How does the concept of limits extend to multivariable functions?

**Tip:** If a limit tends towards infinity, you can use L'Hopital's Rule to simplify some expressions.

Discover how to find the limit of the expression (x + 64 - 8) / x as x approaches 0. This problem explores the concept of limits, algebraic simplification, and the behavior of functions near infinity. Suitable for students in Grades 10-12.

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