The limit to be evaluated is
lim
x→2

x + 47
− 7
x − 2
.

By the direct substitution, we have the following.
lim
x→2

x + 47
− 7
x − 2
=
0
Thus, the direct substitution results in
---Select---
form.

Let's evaluate the limit step by step.

The given limit is:

\[
\lim_{{x \to 2}} \frac{x + 47 - 7}{x - 2}
\]

First, simplify the expression:

\[
\lim_{{x \to 2}} \frac{x + 40}{x - 2}
\]

Now, substitute \( x = 2 \) directly:

\[
\frac{2 + 40}{2 - 2} = \frac{42}{0}
\]

Since the denominator becomes zero and the numerator is a non-zero number (42), the limit results in a form where the expression tends to infinity. This is often referred to as a "division by zero" form, which is not defined and typically leads to a conclusion that the limit does not exist or tends to infinity.

Thus, the direct substitution results in a **undefined** form.

Would you like further details or have any questions?

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Here are 5 related questions you might find helpful:
1. What is the difference between an indeterminate form and an undefined form?
2. How do you handle limits that result in an indeterminate form?
3. Can you evaluate the limit of a function as it approaches infinity?
4. What are the common techniques for finding limits?
5. How does the concept of continuity affect the evaluation of limits?

**Tip:** When direct substitution in a limit results in division by zero, it often indicates that the limit may approach infinity or the limit does not exist. In such cases, further analysis is needed.

Learn how to evaluate the limit of (x + 47) / (x - 2) as x approaches 2 with a detailed step-by-step solution. This problem covers limits, direct substitution, and handling of indeterminate forms in calculus. Suitable for high school students.

Math Problem Statement

x + 47 − 7 x − 2

Solution

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