Calculus Examples

$lim_{x \to 2} \frac{\ln (1.9) - \ln (2)}{1.9 - 2}$

Step 1

Evaluate the limit of $\frac{\ln (1.9) - \ln (2)}{1.9 - 2}$ which is constant as $x$ approaches $2$ .

$\frac{\ln (1.9) - \ln (2)}{1.9 - 2}$

Step 2

Simplify the answer.

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Step 2.1

Simplify the numerator.

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Step 2.1.1

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\frac{\ln (\frac{1.9}{2})}{1.9 - 2}$

Step 2.1.2

Divide $1.9$ by $2$ .

$\frac{\ln (0.95)}{1.9 - 2}$

Step 2.2

Subtract $2$ from $1.9$ .

$\frac{\ln (0.95)}{- 0.1}$

Step 2.3

Move the negative in front of the fraction.

$- \frac{\ln (0.95)}{0.1}$

Step 2.4

Replace $e$ with an approximation.

$- \frac{\log_{2.71828182} (0.95)}{0.1}$

Step 2.5

Log base $2.71828182$ of $0.95$ is approximately $- 0.05129329$ .

$- \frac{- 0.05129329}{0.1}$

Step 2.6

Divide $- 0.05129329$ by $0.1$ .

$- - 0.51293294$

Step 2.7

Multiply $- 1$ by $- 0.51293294$ .

$0.51293294$