Calculus Examples

$lim_{y \to - 3} v (\frac{y^{2} - 4}{2 y^{2} + 7 y + 3})$

Step 1

Set up the limit as a left-sided limit.

$lim_{y \to {(- 3)}^{-}} v (\frac{y^{2} - 4}{2 y^{2} + 7 y + 3})$

Step 2

Evaluate the limits by plugging in the value for the variable.

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Step 2.1

Evaluate the limit of $v (\frac{y^{2} - 4}{2 y^{2} + 7 y + 3})$ by plugging in $- 3$ for $y$ .

$v (\frac{{(- 3)}^{2} - 4}{2 {(- 3)}^{2} + 7 \cdot - 3 + 3})$

Step 2.2

Simplify each term.

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Step 2.2.1

Raise $- 3$ to the power of $2$ .

$v (\frac{{(- 3)}^{2} - 4}{2 \cdot 9 + 7 \cdot - 3 + 3})$

Step 2.2.2

Multiply $2$ by $9$ .

$v (\frac{{(- 3)}^{2} - 4}{18 + 7 \cdot - 3 + 3})$

Step 2.2.3

Multiply $7$ by $- 3$ .

$v (\frac{{(- 3)}^{2} - 4}{18 - 21 + 3})$

Step 2.3

Subtract $21$ from $18$ .

$v (\frac{{(- 3)}^{2} - 4}{- 3 + 3})$

Step 2.4

Add $- 3$ and $3$ .

$v (\frac{{(- 3)}^{2} - 4}{0})$

Step 2.5

Since $v (\frac{{(- 3)}^{2} - 4}{0})$ is undefined, the limit does not exist.

$Does not exist$

Step 3

Set up the limit as a right-sided limit.

$lim_{y \to {(- 3)}^{+}} v (\frac{y^{2} - 4}{2 y^{2} + 7 y + 3})$

Step 4

Evaluate the limits by plugging in the value for the variable.

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Step 4.1

Evaluate the limit of $v (\frac{y^{2} - 4}{2 y^{2} + 7 y + 3})$ by plugging in $- 3$ for $y$ .

$v (\frac{{(- 3)}^{2} - 4}{2 {(- 3)}^{2} + 7 \cdot - 3 + 3})$

Step 4.2

Simplify each term.

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Step 4.2.1

Raise $- 3$ to the power of $2$ .

$v (\frac{{(- 3)}^{2} - 4}{2 \cdot 9 + 7 \cdot - 3 + 3})$

Step 4.2.2

Multiply $2$ by $9$ .

$v (\frac{{(- 3)}^{2} - 4}{18 + 7 \cdot - 3 + 3})$

Step 4.2.3

Multiply $7$ by $- 3$ .

$v (\frac{{(- 3)}^{2} - 4}{18 - 21 + 3})$

Step 4.3

Subtract $21$ from $18$ .

$v (\frac{{(- 3)}^{2} - 4}{- 3 + 3})$

Step 4.4

Add $- 3$ and $3$ .

$v (\frac{{(- 3)}^{2} - 4}{0})$

Step 4.5

Since $v (\frac{{(- 3)}^{2} - 4}{0})$ is undefined, the limit does not exist.

$Does not exist$

Step 5

If either of the one-sided limits does not exist, the limit does not exist.

$Does not exist$