Calculus Examples

$lim_{x \to 10} (90 x^{2}) - 3600 x + 36000$

Step 1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $10$ .

$lim_{x \to 10} 90 x^{2} - lim_{x \to 10} 3600 x + lim_{x \to 10} 36000$

Step 2

Move the term $90$ outside of the limit because it is constant with respect to $x$ .

$90 lim_{x \to 10} x^{2} - lim_{x \to 10} 3600 x + lim_{x \to 10} 36000$

Step 3

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

$90 {(lim_{x \to 10} x)}^{2} - lim_{x \to 10} 3600 x + lim_{x \to 10} 36000$

Step 4

Move the term $3600$ outside of the limit because it is constant with respect to $x$ .

$90 {(lim_{x \to 10} x)}^{2} - 3600 lim_{x \to 10} x + lim_{x \to 10} 36000$

Step 5

Evaluate the limit of $36000$ which is constant as $x$ approaches $10$ .

$90 {(lim_{x \to 10} x)}^{2} - 3600 lim_{x \to 10} x + 36000$

Step 6

Evaluate the limits by plugging in $10$ for all occurrences of $x$ .

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Step 6.1

Evaluate the limit of $x$ by plugging in $10$ for $x$ .

$90 \cdot 10^{2} - 3600 lim_{x \to 10} x + 36000$

Step 6.2

Evaluate the limit of $x$ by plugging in $10$ for $x$ .

$90 \cdot 10^{2} - 3600 \cdot 10 + 36000$

Step 7

Simplify the answer.

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Step 7.1

Raise $10$ to the power of $1$ .

$90 \cdot 10^{2} - 3600 \cdot 10^{1} + 36000$

Step 7.2

Move the decimal point in $- 3600$ to the left by $1$ place and increase the power of $10^{1}$ by $1$ .

$90 \cdot 10^{2} - 360 \cdot 10^{2} + 36000$

Step 7.3

Factor $10^{2}$ out of $90 \cdot 10^{2} - 360 \cdot 10^{2}$ .

$(90 - 360) \cdot 10^{2} + 36000$

Step 7.4

Subtract $360$ from $90$ .

$- 270 \cdot 10^{2} + 36000$

Step 7.5

Move the decimal point in $- 270$ to the left by $2$ places and increase the power of $10^{2}$ by $2$ .

$- 2.7 \cdot 10^{4} + 36000$

Step 7.6

Convert $36000$ to scientific notation.

$- 2.7 \cdot 10^{4} + 3.6 \cdot 10^{4}$

Step 7.7

Factor $10^{4}$ out of $- 2.7 \cdot 10^{4} + 3.6 \cdot 10^{4}$ .

$(- 2.7 + 3.6) \cdot 10^{4}$

Step 7.8

Add $- 2.7$ and $3.6$ .

$0.9 \cdot 10^{4}$

Step 7.9

Move the decimal point in $0.9$ to the right by $1$ place and decrease the power of $10^{4}$ by $1$ .

$9 \cdot 10^{3}$

Step 8

The result can be shown in multiple forms.

Scientific Notation:

$9 \cdot 10^{3}$

Expanded Form:

$9000$