Calculus Examples

$lim_{x \to 2} {(g (x) - 1.5)}^{2}$

Step 1

Evaluate the limit.

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Step 1.1

Move the exponent $2$ from ${(g (x) - 1.5)}^{2}$ outside the limit using the Limits Power Rule.

${(lim_{x \to 2} g (x) - 1.5)}^{2}$

Step 1.2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $2$ .

${(lim_{x \to 2} g (x) - lim_{x \to 2} 1.5)}^{2}$

Step 1.3

Evaluate the limit of $1.5$ which is constant as $x$ approaches $2$ .

${(lim_{x \to 2} g (x) - 1 \cdot 1.5)}^{2}$

Step 2

Evaluate the limit of $g (x)$ by plugging in $2$ for $x$ .

${(g (2) - 1 \cdot 1.5)}^{2}$

Step 3

Simplify the answer.

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Step 3.1

Multiply $- 1$ by $1.5$ .

${(g (2) - 1.5)}^{2}$

Step 3.2

Rewrite ${(g (2) - 1.5)}^{2}$ as $(g (2) - 1.5) (g (2) - 1.5)$ .

$(g (2) - 1.5) (g (2) - 1.5)$

Step 3.3

Expand $(g (2) - 1.5) (g (2) - 1.5)$ using the FOIL Method.

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Step 3.3.1

Apply the distributive property.

$g (2) (g (2) - 1.5) - 1.5 (g (2) - 1.5)$

Step 3.3.2

Apply the distributive property.

$g (2) g (2) + g (2) \cdot - 1.5 - 1.5 (g (2) - 1.5)$

Step 3.3.3

Apply the distributive property.

$g (2) g (2) + g (2) \cdot - 1.5 - 1.5 g (2) - 1.5 \cdot - 1.5$

Step 3.4

Simplify and combine like terms.

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Step 3.4.1

Simplify each term.

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Step 3.4.1.1

Multiply $g (2) g (2)$ .

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Step 3.4.1.1.1

Raise $g (2)$ to the power of $1$ .

${g (2)}^{1} g (2) + g (2) \cdot - 1.5 - 1.5 g (2) - 1.5 \cdot - 1.5$

Step 3.4.1.1.2

Raise $g (2)$ to the power of $1$ .

${g (2)}^{1} {g (2)}^{1} + g (2) \cdot - 1.5 - 1.5 g (2) - 1.5 \cdot - 1.5$

Step 3.4.1.1.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

${g (2)}^{1 + 1} + g (2) \cdot - 1.5 - 1.5 g (2) - 1.5 \cdot - 1.5$

Step 3.4.1.1.4

Add $1$ and $1$ .

${g (2)}^{2} + g (2) \cdot - 1.5 - 1.5 g (2) - 1.5 \cdot - 1.5$

Step 3.4.1.2

Move $- 1.5$ to the left of $g (2)$ .

${g (2)}^{2} - 1.5 \cdot g (2) - 1.5 g (2) - 1.5 \cdot - 1.5$

Step 3.4.1.3

Multiply $- 1.5$ by $- 1.5$ .

${g (2)}^{2} - 1.5 g (2) - 1.5 g (2) + 2.25$

Step 3.4.2

Subtract $1.5 g (2)$ from $- 1.5 g (2)$ .

${g (2)}^{2} - 3 g (2) + 2.25$