Math / Calculus

QuestionFind the limit if it exists. $\lim _{x \rightarrow-2}(7 x+3)$
Which of the following shows the correct expression after the limit properties have been applied? A. $7 \cdot \lim _{x \rightarrow-2} x$ B. 7・ $\lim _{x \rightarrow-2} x \cdot \lim _{x \rightarrow-2} 3$ C. $7 \cdot \lim _{x \rightarrow-2} x+\lim _{x \rightarrow-2} 3$ D. $\lim _{x \rightarrow-2} 3$

Studdy Solution

STEP 1

1. The limit we are evaluating is $\lim_{x \to -2}(7x + 3)$ .
2. The limit of a linear function can be evaluated by direct substitution if the function is continuous at the given point.
3. We can use properties of limits, such as the sum and scalar multiple rules, to simplify the process.

STEP 2

1. Apply the properties of limits to separate the expression into simpler components.
2. Evaluate the limit of each component individually.
3. Combine the results to find the final limit.

STEP 3

Apply the properties of limits to separate the expression. We use the linearity property of limits which states that the limit of a sum is the sum of the limits, and the limit of a constant times a function is the constant times the limit of the function.
$\lim_{x \to -2}(7x + 3) = 7 \cdot \lim_{x \to -2} x + \lim_{x \to -2} 3$

STEP 4

Evaluate the limit of each component. First, evaluate $\lim_{x \to -2} x$ .
$\lim_{x \to -2} x = -2$

STEP 5

Next, evaluate $\lim_{x \to -2} 3$ . Since 3 is a constant, its limit is simply 3.
$\lim_{x \to -2} 3 = 3$

STEP 6

Combine the results obtained from the individual limits. Substitute the evaluated limits back into the separated expression.
$7 \cdot (-2) + 3 = -14 + 3 = -11$

STEP 7

Identify the correct option from the given choices. The expression after applying the limit properties is:
$7 \cdot \lim_{x \to -2} x + \lim_{x \to -2} 3$
This matches option C.
The correct expression after the limit properties have been applied is: C. $7 \cdot \lim_{x \to -2} x + \lim_{x \to -2} 3$

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Studdy Solution

STEP 1

1. The limit we are evaluating is $\lim_{x \to -2}(7x + 3)$ .
2. The limit of a linear function can be evaluated by direct substitution if the function is continuous at the given point.
3. We can use properties of limits, such as the sum and scalar multiple rules, to simplify the process.

STEP 2

1. Apply the properties of limits to separate the expression into simpler components.
2. Evaluate the limit of each component individually.
3. Combine the results to find the final limit.

STEP 3

STEP 4

Evaluate the limit of each component. First, evaluate $\lim_{x \to -2} x$ .
$\lim_{x \to -2} x = -2$

STEP 5

Next, evaluate $\lim_{x \to -2} 3$ . Since 3 is a constant, its limit is simply 3.
$\lim_{x \to -2} 3 = 3$

STEP 6

Combine the results obtained from the individual limits. Substitute the evaluated limits back into the separated expression.
$7 \cdot (-2) + 3 = -14 + 3 = -11$

STEP 7

Identify the correct option from the given choices. The expression after applying the limit properties is:
$7 \cdot \lim_{x \to -2} x + \lim_{x \to -2} 3$
This matches option C.
The correct expression after the limit properties have been applied is: C. $7 \cdot \lim_{x \to -2} x + \lim_{x \to -2} 3$

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Studdy Solution

STEP 1

STEP 2

1. Apply the properties of limits to separate the expression into simpler components.2. Evaluate the limit of each component individually.3. Combine the results to find the final limit.

STEP 3

STEP 4

Evaluate the limit of each component. First, evaluate lim⁡x→−2x\lim_{x \to -2} xlimx→−2​x.lim⁡x→−2x=−2\lim_{x \to -2} x = -2limx→−2​x=−2

STEP 5

Next, evaluate lim⁡x→−23\lim_{x \to -2} 3limx→−2​3. Since 3 is a constant, its limit is simply 3.lim⁡x→−23=3\lim_{x \to -2} 3 = 3limx→−2​3=3

STEP 6

Combine the results obtained from the individual limits. Substitute the evaluated limits back into the separated expression.7⋅(−2)+3=−14+3=−117 \cdot (-2) + 3 = -14 + 3 = -117⋅(−2)+3=−14+3=−11

STEP 7

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1. Apply the properties of limits to separate the expression into simpler components.
2. Evaluate the limit of each component individually.
3. Combine the results to find the final limit.

Evaluate the limit of each component. First, evaluate $\lim_{x \to -2} x$ .
$\lim_{x \to -2} x = -2$

Next, evaluate $\lim_{x \to -2} 3$ . Since 3 is a constant, its limit is simply 3.
$\lim_{x \to -2} 3 = 3$

Combine the results obtained from the individual limits. Substitute the evaluated limits back into the separated expression.
$7 \cdot (-2) + 3 = -14 + 3 = -11$