Question

For real number a, which of the following equations are true? Select all that apply.

(2 points)
Responses

lim underscript x left arrow a baseline end scripts x equals a
Image with alt text: lim underscript x left arrow a baseline end scripts x equals a

lim underscript x left arrow a baseline end scripts a equals a
Image with alt text: lim underscript x left arrow a baseline end scripts a equals a

lim underscript x left arrow 5 baseline end scripts 4 equals 5
Image with alt text: lim underscript x left arrow 5 baseline end scripts 4 equals 5

lim underscript x left arrow 5 baseline end scripts x equals 5
Image with alt text: lim underscript x left arrow 5 baseline end scripts x equals 5

lim underscript x left arrow a baseline end scripts a equals x

Find limit underscript x right arrow 3 endscript left parenthesis x squared plus 7 right parenthesis.
(1 point)
Responses

9
9

12
12

13
13

16

Answer 1

Let's evaluate each equation in the first part to determine which ones are true.

1. \( \lim_{x \to a} x = a \)

**True.** As \( x \) approaches \( a \), it gets arbitrarily close to \( a \).

2. \( \lim_{x \to a} a = a \)

**True.** Since \( a \) is a constant value, the limit of \( a \) as \( x \) approaches \( a \) is simply \( a \).

3. \( \lim_{x \to 5} 4 = 5 \)

**False.** The limit of a constant (4) is always equal to that constant, regardless of \( x \). So, \( \lim_{x \to 5} 4 = 4 \), not 5.

4. \( \lim_{x \to 5} x = 5 \)

**True.** As \( x \) approaches 5, the value of \( x \) approaches 5.

5. \( \lim_{x \to a} a = x \)

**False.** The limit of \( a \) as \( x \) approaches \( a \) does not equal \( x \); it equals \( a \).

So, the correct responses are:
- \( \lim_{x \to a} x = a \)
- \( \lim_{x \to a} a = a \)
- \( \lim_{x \to 5} x = 5 \)

---

Now, let's find the limit \( \lim_{x \to 3} (x^2 + 7) \).

We can directly substitute \( x = 3 \):

\[
\lim_{x \to 3} (x^2 + 7) = 3^2 + 7 = 9 + 7 = 16
\]

Therefore, the answer is:

**16**