Calculus Examples

$lim_{x \to 999999} \ln (1 + x^{2}) - \ln (1 + x)$

Step 1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $999999$ .

$lim_{x \to 999999} \ln (1 + x^{2}) - lim_{x \to 999999} \ln (1 + x)$

Step 2

Move the limit inside the logarithm.

$\ln (lim_{x \to 999999} 1 + x^{2}) - lim_{x \to 999999} \ln (1 + x)$

Step 3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $999999$ .

$\ln (lim_{x \to 999999} 1 + lim_{x \to 999999} x^{2}) - lim_{x \to 999999} \ln (1 + x)$

Step 4

Evaluate the limit of $1$ which is constant as $x$ approaches $999999$ .

$\ln (1 + lim_{x \to 999999} x^{2}) - lim_{x \to 999999} \ln (1 + x)$

Step 5

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$\ln (1 + {(lim_{x \to 999999} x)}^{2}) - lim_{x \to 999999} \ln (1 + x)$

Step 6

Move the limit inside the logarithm.

$\ln (1 + {(lim_{x \to 999999} x)}^{2}) - \ln (lim_{x \to 999999} 1 + x)$

Step 7

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $999999$ .

$\ln (1 + {(lim_{x \to 999999} x)}^{2}) - \ln (lim_{x \to 999999} 1 + lim_{x \to 999999} x)$

Step 8

Evaluate the limit of $1$ which is constant as $x$ approaches $999999$ .

$\ln (1 + {(lim_{x \to 999999} x)}^{2}) - \ln (1 + lim_{x \to 999999} x)$

Step 9

Evaluate the limits by plugging in $999999$ for all occurrences of $x$ .

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Step 9.1

Evaluate the limit of $x$ by plugging in $999999$ for $x$ .

$\ln (1 + 999999^{2}) - \ln (1 + lim_{x \to 999999} x)$

Step 9.2

Evaluate the limit of $x$ by plugging in $999999$ for $x$ .

$\ln (1 + 999999^{2}) - \ln (1 + 999999)$

Step 10

Simplify the answer.

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Step 10.1

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

$\ln (\frac{1 + 999999^{2}}{1 + 999999})$

Step 10.2

Simplify the numerator.

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Step 10.2.1

Raise $999999$ to the power of $2$ .

$\ln (\frac{1 + 999998000001}{1 + 999999})$

Step 10.2.2

Add $1$ and $999998000001$ .

$\ln (\frac{999998000002}{1 + 999999})$

Step 10.3

Add $1$ and $999999$ .

$\ln (\frac{999998000002}{1000000})$

Step 10.4

Cancel the common factor of $999998000002$ and $1000000$ .

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Step 10.4.1

Factor $2$ out of $999998000002$ .

$\ln (\frac{2 (499999000001)}{1000000})$

Step 10.4.2

Cancel the common factors.

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Step 10.4.2.1

Factor $2$ out of $1000000$ .

$\ln (\frac{2 \cdot 499999000001}{2 \cdot 500000})$

Step 10.4.2.2

Cancel the common factor.

$\ln (\frac{2 \cdot 499999000001}{2 \cdot 500000})$

Step 10.4.2.3

Rewrite the expression.

$\ln (\frac{499999000001}{500000})$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\ln (\frac{499999000001}{500000})$

Decimal Form:

$13.81550855 \dots$