Math / Calculus

QuestionEvaluate the following limit. Give an exact answer if the limit is a number. Otherwise, enter $-\infty$ or $\infty$ if the limit is infinite, or enter DNE if the limit does not exist in another way. $\lim _{x \rightarrow 8} \frac{\sqrt{x+17}-5}{x-8}=$ $\square$
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Studdy Solution

STEP 1

1. The limit in question involves an expression with a square root in the numerator and a linear term in the denominator.
2. Direct substitution of $x=8$ leads to an indeterminate form of type $\frac{0}{0}$ .
3. We can apply algebraic techniques or L'Hôpital's rule to resolve the indeterminate form.

STEP 2

1. Verify the indeterminate form by direct substitution.
2. Apply algebraic manipulation techniques to simplify the expression.
3. Alternatively, apply L'Hôpital's rule to evaluate the limit.
4. Simplify the resulting expression to find the limit.

STEP 3

Verify the indeterminate form by direct substitution of $x = 8$ into the given limit expression.
$\lim_{x \rightarrow 8} \frac{\sqrt{x+17}-5}{x-8}$
Substituting $x = 8$ : $\frac{\sqrt{8+17}-5}{8-8} = \frac{\sqrt{25}-5}{0} = \frac{5-5}{0} = \frac{0}{0}$
The expression is of the form $\frac{0}{0}$ , which is indeterminate.

STEP 4

Use algebraic manipulation, specifically multiplying by the conjugate, to simplify the expression.
Multiply numerator and denominator by the conjugate of the numerator: $\frac{\sqrt{x+17}-5}{x-8} \cdot \frac{\sqrt{x+17}+5}{\sqrt{x+17}+5} = \frac{(\sqrt{x+17}-5)(\sqrt{x+17}+5)}{(x-8)(\sqrt{x+17}+5)}$

STEP 5

Simplify the numerator using the difference of squares formula: $(a-b)(a+b) = a^2 - b^2$ .
$(\sqrt{x+17})^2 - 5^2 = (x+17) - 25 = x - 8$

STEP 6

Substitute the simplified numerator back into the expression.
$\frac{x-8}{(x-8)(\sqrt{x+17}+5)}$

STEP 7

Cancel the common factor $(x-8)$ in the numerator and denominator.
$\frac{1}{\sqrt{x+17}+5}$

STEP 8

Evaluate the limit as $x$ approaches 8.
$\lim_{x \rightarrow 8} \frac{1}{\sqrt{x+17}+5} = \frac{1}{\sqrt{8+17}+5} = \frac{1}{\sqrt{25}+5} = \frac{1}{5+5} = \frac{1}{10}$
The solution to the limit is:
$\boxed{\frac{1}{10}}$

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QuestionEvaluate the following limit. Give an exact answer if the limit is a number. Otherwise, enter $-\infty$ or $\infty$ if the limit is infinite, or enter DNE if the limit does not exist in another way. $\lim _{x \rightarrow 8} \frac{\sqrt{x+17}-5}{x-8}=$ $\square$
Submit answer Next item

Studdy Solution

STEP 1

STEP 2

1. Verify the indeterminate form by direct substitution.
2. Apply algebraic manipulation techniques to simplify the expression.
3. Alternatively, apply L'Hôpital's rule to evaluate the limit.
4. Simplify the resulting expression to find the limit.

STEP 3

STEP 4

Use algebraic manipulation, specifically multiplying by the conjugate, to simplify the expression.
Multiply numerator and denominator by the conjugate of the numerator: $\frac{\sqrt{x+17}-5}{x-8} \cdot \frac{\sqrt{x+17}+5}{\sqrt{x+17}+5} = \frac{(\sqrt{x+17}-5)(\sqrt{x+17}+5)}{(x-8)(\sqrt{x+17}+5)}$

STEP 5

Simplify the numerator using the difference of squares formula: $(a-b)(a+b) = a^2 - b^2$ .
$(\sqrt{x+17})^2 - 5^2 = (x+17) - 25 = x - 8$

STEP 6

Substitute the simplified numerator back into the expression.
$\frac{x-8}{(x-8)(\sqrt{x+17}+5)}$

STEP 7

Cancel the common factor $(x-8)$ in the numerator and denominator.
$\frac{1}{\sqrt{x+17}+5}$

STEP 8

Evaluate the limit as $x$ approaches 8.
$\lim_{x \rightarrow 8} \frac{1}{\sqrt{x+17}+5} = \frac{1}{\sqrt{8+17}+5} = \frac{1}{\sqrt{25}+5} = \frac{1}{5+5} = \frac{1}{10}$
The solution to the limit is:
$\boxed{\frac{1}{10}}$

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Studdy Solution

STEP 1

STEP 2

1. Verify the indeterminate form by direct substitution.2. Apply algebraic manipulation techniques to simplify the expression.3. Alternatively, apply L'Hôpital's rule to evaluate the limit.4. Simplify the resulting expression to find the limit.

STEP 3

STEP 4

STEP 5

Simplify the numerator using the difference of squares formula: (a−b)(a+b)=a2−b2(a-b)(a+b) = a^2 - b^2(a−b)(a+b)=a2−b2.(x+17)2−52=(x+17)−25=x−8 (\sqrt{x+17})^2 - 5^2 = (x+17) - 25 = x - 8 (x+17​)2−52=(x+17)−25=x−8

STEP 6

Substitute the simplified numerator back into the expression.x−8(x−8)(x+17+5) \frac{x-8}{(x-8)(\sqrt{x+17}+5)} (x−8)(x+17​+5)x−8​

STEP 7

Cancel the common factor (x−8)(x-8)(x−8) in the numerator and denominator.1x+17+5 \frac{1}{\sqrt{x+17}+5} x+17​+51​

STEP 8

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1. Verify the indeterminate form by direct substitution.
2. Apply algebraic manipulation techniques to simplify the expression.
3. Alternatively, apply L'Hôpital's rule to evaluate the limit.
4. Simplify the resulting expression to find the limit.

Simplify the numerator using the difference of squares formula: $(a-b)(a+b) = a^2 - b^2$ .
$(\sqrt{x+17})^2 - 5^2 = (x+17) - 25 = x - 8$

Substitute the simplified numerator back into the expression.
$\frac{x-8}{(x-8)(\sqrt{x+17}+5)}$

Cancel the common factor $(x-8)$ in the numerator and denominator.
$\frac{1}{\sqrt{x+17}+5}$