Calculus Examples

$\frac{4000}{\ln (x + 5)}$

Step 1

Find the asymptotes.

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Step 1.1

Find where the expression $\frac{4000}{\ln (x + 5)}$ is undefined.

$x \leq - 5, x = - 4$

Step 1.2

Since $\frac{4000}{\ln (x + 5)}$ $\to$ $- \infty$ as $x$ $\to$ $- 4$ from the left and $\frac{4000}{\ln (x + 5)}$ $\to$ $\infty$ as $x$ $\to$ $- 4$ from the right, then $x = - 4$ is a vertical asymptote.

$x = - 4$

Step 1.3

Evaluate $lim_{x \to \infty} \frac{4000}{\ln (x + 5)}$ to find the horizontal asymptote.

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Step 1.3.1

Move the term $4000$ outside of the limit because it is constant with respect to $x$ .

$4000 lim_{x \to \infty} \frac{1}{\ln (x + 5)}$

Step 1.3.2

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{\ln (x + 5)}$ approaches $0$ .

$4000 \cdot 0$

Step 1.3.3

Multiply $4000$ by $0$ .

$0$

Step 1.4

List the horizontal asymptotes:

$y = 0$

Step 1.5

No oblique asymptotes are present for logarithmic and trigonometric functions.

No Oblique Asymptotes

Step 1.6

This is the set of all asymptotes.

Vertical Asymptotes: $x = - 4$

Horizontal Asymptotes: $y = 0$

Vertical Asymptotes: $x = - 4$

Horizontal Asymptotes: $y = 0$

Step 2

Find the point at $x = - 3$ .

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Step 2.1

Replace the variable $x$ with $- 3$ in the expression.

$f (- 3) = \frac{4000}{\ln ((- 3) + 5)}$

Step 2.2

Simplify the result.

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Step 2.2.1

Add $- 3$ and $5$ .

$f (- 3) = \frac{4000}{\ln (2)}$

Step 2.2.2

The final answer is $\frac{4000}{\ln (2)}$ .

$\frac{4000}{\ln (2)}$

Step 2.3

Convert $\frac{4000}{\ln (2)}$ to decimal.

$y = 5770.78016355$

Step 3

Find the point at $x = - 2$ .

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Step 3.1

Replace the variable $x$ with $- 2$ in the expression.

$f (- 2) = \frac{4000}{\ln ((- 2) + 5)}$

Step 3.2

Simplify the result.

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Step 3.2.1

Add $- 2$ and $5$ .

$f (- 2) = \frac{4000}{\ln (3)}$

Step 3.2.2

The final answer is $\frac{4000}{\ln (3)}$ .

$\frac{4000}{\ln (3)}$

Step 3.3

Convert $\frac{4000}{\ln (3)}$ to decimal.

$y = 3640.9569065$

Step 4

Find the point at $x = - 1$ .

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Step 4.1

Replace the variable $x$ with $- 1$ in the expression.

$f (- 1) = \frac{4000}{\ln ((- 1) + 5)}$

Step 4.2

Simplify the result.

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Step 4.2.1

Add $- 1$ and $5$ .

$f (- 1) = \frac{4000}{\ln (4)}$

Step 4.2.2

Rewrite $\ln (4)$ as $\ln (2^{2})$ .

$f (- 1) = \frac{4000}{\ln (2^{2})}$

Step 4.2.3

Expand $\ln (2^{2})$ by moving $2$ outside the logarithm.

$f (- 1) = \frac{4000}{2 \ln (2)}$

Step 4.2.4

Cancel the common factor of $4000$ and $2$ .

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Step 4.2.4.1

Factor $2$ out of $4000$ .

$f (- 1) = \frac{2 \cdot 2000}{2 \ln (2)}$

Step 4.2.4.2

Cancel the common factors.

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Step 4.2.4.2.1

Factor $2$ out of $2 \ln (2)$ .

$f (- 1) = \frac{2 \cdot 2000}{2 (\ln (2))}$

Step 4.2.4.2.2

Cancel the common factor.

$f (- 1) = \frac{2 \cdot 2000}{2 \ln (2)}$

Step 4.2.4.2.3

Rewrite the expression.

$f (- 1) = \frac{2000}{\ln (2)}$

Step 4.2.5

The final answer is $\frac{2000}{\ln (2)}$ .

$\frac{2000}{\ln (2)}$

Step 4.3

Convert $\frac{2000}{\ln (2)}$ to decimal.

$y = 2885.39008177$

Step 5

The log function can be graphed using the vertical asymptote at $x = - 4$ and the points $(- 3, 5770.78016355), (- 2, 3640.9569065), (- 1, 2885.39008177)$ .

Vertical Asymptote: $x = - 4$

$\begin{array}{c} x & y \\ - 3 & 5770.78 \\ - 2 & 3640.957 \\ - 1 & 2885.39 \end{array}$

Step 6