Calculus Examples

$lim_{h \to 0} ({(a + h)}^{2} - 3 (a + h) + 3) - {(a)}^{2} + 3 (a) - 3$

Step 1

Split the limit using the Sum of Limits Rule on the limit as $h$ approaches $0$ .

$lim_{h \to 0} {(a + h)}^{2} - lim_{h \to 0} 3 (a + h) + lim_{h \to 0} 3 - lim_{h \to 0} {(a)}^{2} + lim_{h \to 0} 3 (a) - lim_{h \to 0} 3$

Step 2

Move the exponent $2$ from ${(a + h)}^{2}$ outside the limit using the Limits Power Rule.

${(lim_{h \to 0} a + h)}^{2} - lim_{h \to 0} 3 (a + h) + lim_{h \to 0} 3 - lim_{h \to 0} {(a)}^{2} + lim_{h \to 0} 3 (a) - lim_{h \to 0} 3$

Step 3

Split the limit using the Sum of Limits Rule on the limit as $h$ approaches $0$ .

${(lim_{h \to 0} a + lim_{h \to 0} h)}^{2} - lim_{h \to 0} 3 (a + h) + lim_{h \to 0} 3 - lim_{h \to 0} {(a)}^{2} + lim_{h \to 0} 3 (a) - lim_{h \to 0} 3$

Step 4

Evaluate the limit of $a$ which is constant as $h$ approaches $0$ .

${(a + lim_{h \to 0} h)}^{2} - lim_{h \to 0} 3 (a + h) + lim_{h \to 0} 3 - lim_{h \to 0} {(a)}^{2} + lim_{h \to 0} 3 (a) - lim_{h \to 0} 3$

Step 5

Move the term $3$ outside of the limit because it is constant with respect to $h$ .

${(a + lim_{h \to 0} h)}^{2} - 3 lim_{h \to 0} a + h + lim_{h \to 0} 3 - lim_{h \to 0} {(a)}^{2} + lim_{h \to 0} 3 (a) - lim_{h \to 0} 3$

Step 6

Split the limit using the Sum of Limits Rule on the limit as $h$ approaches $0$ .

${(a + lim_{h \to 0} h)}^{2} - 3 (lim_{h \to 0} a + lim_{h \to 0} h) + lim_{h \to 0} 3 - lim_{h \to 0} {(a)}^{2} + lim_{h \to 0} 3 (a) - lim_{h \to 0} 3$

Step 7

Evaluate the limit of $a$ which is constant as $h$ approaches $0$ .

${(a + lim_{h \to 0} h)}^{2} - 3 (a + lim_{h \to 0} h) + lim_{h \to 0} 3 - lim_{h \to 0} {(a)}^{2} + lim_{h \to 0} 3 (a) - lim_{h \to 0} 3$

Step 8

Evaluate the limit of $3$ which is constant as $h$ approaches $0$ .

${(a + lim_{h \to 0} h)}^{2} - 3 (a + lim_{h \to 0} h) + 3 - lim_{h \to 0} {(a)}^{2} + lim_{h \to 0} 3 (a) - lim_{h \to 0} 3$

Step 9

Evaluate the limit of ${(a)}^{2}$ which is constant as $h$ approaches $0$ .

${(a + lim_{h \to 0} h)}^{2} - 3 (a + lim_{h \to 0} h) + 3 - {(a)}^{2} + lim_{h \to 0} 3 (a) - lim_{h \to 0} 3$

Step 10

Evaluate the limit of $3 (a)$ which is constant as $h$ approaches $0$ .

${(a + lim_{h \to 0} h)}^{2} - 3 (a + lim_{h \to 0} h) + 3 - {(a)}^{2} + 3 (a) - lim_{h \to 0} 3$

Step 11

Evaluate the limit of $3$ which is constant as $h$ approaches $0$ .

${(a + lim_{h \to 0} h)}^{2} - 3 (a + lim_{h \to 0} h) + 3 - {(a)}^{2} + 3 (a) - 1 \cdot 3$

Step 12

Evaluate the limits by plugging in $0$ for all occurrences of $h$ .

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Step 12.1

Evaluate the limit of $h$ by plugging in $0$ for $h$ .

${(a + 0)}^{2} - 3 (a + lim_{h \to 0} h) + 3 - {(a)}^{2} + 3 (a) - 1 \cdot 3$

Step 12.2

Evaluate the limit of $h$ by plugging in $0$ for $h$ .

${(a + 0)}^{2} - 3 (a + 0) + 3 - {(a)}^{2} + 3 (a) - 1 \cdot 3$

Step 13

Simplify the answer.

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Step 13.1

Combine the opposite terms in ${(a + 0)}^{2} - 3 (a + 0) + 3 - {(a)}^{2} + 3 (a) - 1 \cdot 3$ .

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Step 13.1.1

Add $a$ and $0$ .

$a^{2} - 3 (a + 0) + 3 - {(a)}^{2} + 3 (a) - 1 \cdot 3$

Step 13.1.2

Add $a$ and $0$ .

$a^{2} - 3 a + 3 - {(a)}^{2} + 3 (a) - 1 \cdot 3$

Step 13.1.3

Add $- 3 a$ and $3 (a)$ .

$a^{2} + 3 - {(a)}^{2} + 0 - 1 \cdot 3$

Step 13.1.4

Add $a^{2} + 3 - {(a)}^{2}$ and $0$ .

$a^{2} + 3 - {(a)}^{2} - 1 \cdot 3$

Step 13.2

Multiply $- 1$ by $3$ .

$a^{2} + 3 - a^{2} - 3$

Step 13.3

Combine the opposite terms in $a^{2} + 3 - a^{2} - 3$ .

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Step 13.3.1

Subtract $a^{2}$ from $a^{2}$ .

$0 + 3 - 3$

Step 13.3.2

Add $0$ and $3$ .

$3 - 3$

Step 13.3.3

Subtract $3$ from $3$ .

$0$