Math / Calculus

Question2) $\lim _{x \rightarrow 0} \frac{(x+3)^{2}-9}{x}$

Studdy Solution

STEP 1

1. We are given the limit expression $\lim _{x \rightarrow 0} \frac{(x+3)^{2}-9}{x}$ .
2. The expression involves a rational function where the numerator is a difference of squares.
3. We need to evaluate the limit as $x$ approaches 0.

STEP 2

1. Simplify the numerator using algebraic identities.
2. Factor the expression to cancel terms.
3. Evaluate the limit after simplification.

STEP 3

Recognize that the numerator $(x+3)^{2} - 9$ is a difference of squares. Use the identity $a^2 - b^2 = (a-b)(a+b)$ .
$(x+3)^2 - 9 = ((x+3) - 3)((x+3) + 3)$

STEP 4

Simplify the expression:
$(x+3)^2 - 9 = (x)(x+6)$
Substitute back into the limit expression:
$\lim _{x \rightarrow 0} \frac{x(x+6)}{x}$

STEP 5

Cancel the common factor $x$ in the numerator and the denominator:
$\lim _{x \rightarrow 0} (x+6)$

STEP 6

Evaluate the limit as $x$ approaches 0:
$\lim _{x \rightarrow 0} (x+6) = 0 + 6 = 6$
The value of the limit is:
$\boxed{6}$

Was this helpful?

Question2) $\lim _{x \rightarrow 0} \frac{(x+3)^{2}-9}{x}$

Studdy Solution

STEP 1

1. We are given the limit expression $\lim _{x \rightarrow 0} \frac{(x+3)^{2}-9}{x}$ .
2. The expression involves a rational function where the numerator is a difference of squares.
3. We need to evaluate the limit as $x$ approaches 0.

STEP 2

1. Simplify the numerator using algebraic identities.
2. Factor the expression to cancel terms.
3. Evaluate the limit after simplification.

STEP 3

Recognize that the numerator $(x+3)^{2} - 9$ is a difference of squares. Use the identity $a^2 - b^2 = (a-b)(a+b)$ .
$(x+3)^2 - 9 = ((x+3) - 3)((x+3) + 3)$

STEP 4

Simplify the expression:
$(x+3)^2 - 9 = (x)(x+6)$
Substitute back into the limit expression:
$\lim _{x \rightarrow 0} \frac{x(x+6)}{x}$

STEP 5

Cancel the common factor $x$ in the numerator and the denominator:
$\lim _{x \rightarrow 0} (x+6)$

STEP 6

Evaluate the limit as $x$ approaches 0:
$\lim _{x \rightarrow 0} (x+6) = 0 + 6 = 6$
The value of the limit is:
$\boxed{6}$

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Question2) lim⁡x→0(x+3)2−9x\lim _{x \rightarrow 0} \frac{(x+3)^{2}-9}{x}limx→0​x(x+3)2−9​

Studdy Solution

STEP 1

1. We are given the limit expression lim⁡x→0(x+3)2−9x\lim _{x \rightarrow 0} \frac{(x+3)^{2}-9}{x}limx→0​x(x+3)2−9​.2. The expression involves a rational function where the numerator is a difference of squares.3. We need to evaluate the limit as x x x approaches 0.

STEP 2

1. Simplify the numerator using algebraic identities.2. Factor the expression to cancel terms.3. Evaluate the limit after simplification.

STEP 3

Recognize that the numerator (x+3)2−9(x+3)^{2} - 9(x+3)2−9 is a difference of squares. Use the identity a2−b2=(a−b)(a+b)a^2 - b^2 = (a-b)(a+b)a2−b2=(a−b)(a+b).(x+3)2−9=((x+3)−3)((x+3)+3) (x+3)^2 - 9 = ((x+3) - 3)((x+3) + 3) (x+3)2−9=((x+3)−3)((x+3)+3)

STEP 4

Simplify the expression:(x+3)2−9=(x)(x+6) (x+3)^2 - 9 = (x)(x+6) (x+3)2−9=(x)(x+6)Substitute back into the limit expression:lim⁡x→0x(x+6)x \lim _{x \rightarrow 0} \frac{x(x+6)}{x} x→0lim​xx(x+6)​

STEP 5

Cancel the common factor xxx in the numerator and the denominator:lim⁡x→0(x+6) \lim _{x \rightarrow 0} (x+6) x→0lim​(x+6)

STEP 6

Evaluate the limit as xxx approaches 0:lim⁡x→0(x+6)=0+6=6 \lim _{x \rightarrow 0} (x+6) = 0 + 6 = 6 x→0lim​(x+6)=0+6=6 The value of the limit is:6 \boxed{6} 6​

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Question2) $\lim _{x \rightarrow 0} \frac{(x+3)^{2}-9}{x}$

1. We are given the limit expression $\lim _{x \rightarrow 0} \frac{(x+3)^{2}-9}{x}$ .
2. The expression involves a rational function where the numerator is a difference of squares.
3. We need to evaluate the limit as $x$ approaches 0.

1. Simplify the numerator using algebraic identities.
2. Factor the expression to cancel terms.
3. Evaluate the limit after simplification.

Recognize that the numerator $(x+3)^{2} - 9$ is a difference of squares. Use the identity $a^2 - b^2 = (a-b)(a+b)$ .
$(x+3)^2 - 9 = ((x+3) - 3)((x+3) + 3)$

Simplify the expression:
$(x+3)^2 - 9 = (x)(x+6)$
Substitute back into the limit expression:
$\lim _{x \rightarrow 0} \frac{x(x+6)}{x}$

Cancel the common factor $x$ in the numerator and the denominator:
$\lim _{x \rightarrow 0} (x+6)$

Evaluate the limit as $x$ approaches 0:
$\lim _{x \rightarrow 0} (x+6) = 0 + 6 = 6$
The value of the limit is:
$\boxed{6}$