Question

For g left parenthesis x right parenthesis equals 5 and h left parenthesis x right parenthesis equals 2x, find "lim underscript x left arrow a endscript left parenthesis g left parenthesis x right parenthesis times h left parenthesis x right parenthesis right parenthesis".
(1 point)
Responses

3a
3 a

10a
10 a

2a squared
Image with alt text: 2a squared

2a + 5

For g left parenthesis x right parenthesis equals x minus 10 and h left parenthesis x right parenthesis equals 2x, find lim underscript x left arrow negative 2 baseline endscripts left parenthesis start fraction g left parenthesis x right parenthesis over h left parenthesis x right parenthesis end fraction right parenthesis.
(1 point)
Responses

negative 3
Image with alt text: negative 3

negative 2
Image with alt text: negative 2

2
2

3

Answer 1

Let's solve each limit step by step:

1. For the first limit:

Given:
- \( g(x) = 5 \)
- \( h(x) = 2x \)

We need to find:
\[
\lim_{x \to a} (g(x) \cdot h(x))
\]

We can evaluate this limit as follows:
\[
g(x) \cdot h(x) = 5 \cdot (2x) = 10x
\]

Now we find the limit as \( x \) approaches \( a \):
\[
\lim_{x \to a} (10x) = 10a
\]

Therefore, the answer for the first limit is:
\[
\boxed{10a}
\]

2. For the second limit:

Given:
- \( g(x) = x - 10 \)
- \( h(x) = 2x \)

We need to find:
\[
\lim_{x \to -2} \left( \frac{g(x)}{h(x)} \right)
\]

Let's evaluate \( g(-2) \) and \( h(-2) \):
\[
g(-2) = -2 - 10 = -12
\]
\[
h(-2) = 2(-2) = -4
\]

Now we compute the limit:
\[
\lim_{x \to -2} \left( \frac{g(x)}{h(x)} \right) = \frac{g(-2)}{h(-2)} = \frac{-12}{-4} = 3
\]

Therefore, the answer for the second limit is:
\[
\boxed{3}
\]