Calculus Examples

$lim_{x \to 5} 3 x^{2} - 4 x - 1$

Step 1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $5$ .

$lim_{x \to 5} 3 x^{2} - lim_{x \to 5} 4 x - lim_{x \to 5} 1$

Step 2

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

$3 lim_{x \to 5} x^{2} - lim_{x \to 5} 4 x - lim_{x \to 5} 1$

Step 3

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

$3 {(lim_{x \to 5} x)}^{2} - lim_{x \to 5} 4 x - lim_{x \to 5} 1$

Step 4

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$3 {(lim_{x \to 5} x)}^{2} - 4 lim_{x \to 5} x - lim_{x \to 5} 1$

Step 5

Evaluate the limit of $1$ which is constant as $x$ approaches $5$ .

$3 {(lim_{x \to 5} x)}^{2} - 4 lim_{x \to 5} x - 1 \cdot 1$

Step 6

Evaluate the limits by plugging in $5$ for all occurrences of $x$ .

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Step 6.1

Evaluate the limit of $x$ by plugging in $5$ for $x$ .

$3 \cdot 5^{2} - 4 lim_{x \to 5} x - 1 \cdot 1$

Step 6.2

Evaluate the limit of $x$ by plugging in $5$ for $x$ .

$3 \cdot 5^{2} - 4 \cdot 5 - 1 \cdot 1$

Step 7

Simplify the answer.

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Step 7.1

Simplify each term.

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Step 7.1.1

Raise $5$ to the power of $2$ .

$3 \cdot 25 - 4 \cdot 5 - 1 \cdot 1$

Step 7.1.2

Multiply $3$ by $25$ .

$75 - 4 \cdot 5 - 1 \cdot 1$

Step 7.1.3

Multiply $- 4$ by $5$ .

$75 - 20 - 1 \cdot 1$

Step 7.1.4

Multiply $- 1$ by $1$ .

$75 - 20 - 1$

Step 7.2

Subtract $20$ from $75$ .

$55 - 1$

Step 7.3

Subtract $1$ from $55$ .

$54$