Calculus Examples

$lim_{x \to 1} \frac{x^{3} + 2 x^{2} + 3 x}{x}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $1$ .

$\frac{lim_{x \to 1} x^{3} + 2 x^{2} + 3 x}{lim_{x \to 1} x}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $1$ .

$\frac{lim_{x \to 1} x^{3} + lim_{x \to 1} 2 x^{2} + lim_{x \to 1} 3 x}{lim_{x \to 1} x}$

Step 3

Move the exponent $3$ from $x^{3}$ outside the limit using the Limits Power Rule.

$\frac{{(lim_{x \to 1} x)}^{3} + lim_{x \to 1} 2 x^{2} + lim_{x \to 1} 3 x}{lim_{x \to 1} x}$

Step 4

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{{(lim_{x \to 1} x)}^{3} + 2 lim_{x \to 1} x^{2} + lim_{x \to 1} 3 x}{lim_{x \to 1} x}$

Step 5

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

$\frac{{(lim_{x \to 1} x)}^{3} + 2 {(lim_{x \to 1} x)}^{2} + lim_{x \to 1} 3 x}{lim_{x \to 1} x}$

Step 6

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

$\frac{{(lim_{x \to 1} x)}^{3} + 2 {(lim_{x \to 1} x)}^{2} + 3 lim_{x \to 1} x}{lim_{x \to 1} x}$

Step 7

Evaluate the limits by plugging in $1$ for all occurrences of $x$ .

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Step 7.1

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$\frac{1^{3} + 2 {(lim_{x \to 1} x)}^{2} + 3 lim_{x \to 1} x}{lim_{x \to 1} x}$

Step 7.2

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$\frac{1^{3} + 2 \cdot 1^{2} + 3 lim_{x \to 1} x}{lim_{x \to 1} x}$

Step 7.3

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$\frac{1^{3} + 2 \cdot 1^{2} + 3 \cdot 1}{lim_{x \to 1} x}$

Step 7.4

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$\frac{1^{3} + 2 \cdot 1^{2} + 3 \cdot 1}{1}$

Step 8

Simplify the answer.

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Step 8.1

Divide $1^{3} + 2 \cdot 1^{2} + 3 \cdot 1$ by $1$ .

$1^{3} + 2 \cdot 1^{2} + 3 \cdot 1$

Step 8.2

Simplify each term.

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Step 8.2.1

One to any power is one.

$1 + 2 \cdot 1^{2} + 3 \cdot 1$

Step 8.2.2

One to any power is one.

$1 + 2 \cdot 1 + 3 \cdot 1$

Step 8.2.3

Multiply $2$ by $1$ .

$1 + 2 + 3 \cdot 1$

Step 8.2.4

Multiply $3$ by $1$ .

$1 + 2 + 3$

Step 8.3

Add $1$ and $2$ .

$3 + 3$

Step 8.4

Add $3$ and $3$ .

$6$