Calculus Examples

$lim_{y \to 2} y^{2} - \frac{1}{y}$

Step 1

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

$lim_{y \to 2} y^{2} - lim_{y \to 2} \frac{1}{y}$

Step 2

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

${(lim_{y \to 2} y)}^{2} - lim_{y \to 2} \frac{1}{y}$

Step 3

Split the limit using the Limits Quotient Rule on the limit as $y$ approaches $2$ .

${(lim_{y \to 2} y)}^{2} - \frac{lim_{y \to 2} 1}{lim_{y \to 2} y}$

Step 4

Evaluate the limit of $1$ which is constant as $y$ approaches $2$ .

${(lim_{y \to 2} y)}^{2} - \frac{1}{lim_{y \to 2} y}$

Step 5

Evaluate the limits by plugging in $2$ for all occurrences of $y$ .

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Step 5.1

Evaluate the limit of $y$ by plugging in $2$ for $y$ .

$2^{2} - \frac{1}{lim_{y \to 2} y}$

Step 5.2

Evaluate the limit of $y$ by plugging in $2$ for $y$ .

$2^{2} - \frac{1}{2}$

Step 6

Simplify the answer.

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Step 6.1

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

$4 - \frac{1}{2}$

Step 6.2

To write $4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$4 \cdot \frac{2}{2} - \frac{1}{2}$

Step 6.3

Combine $4$ and $\frac{2}{2}$ .

$\frac{4 \cdot 2}{2} - \frac{1}{2}$

Step 6.4

Combine the numerators over the common denominator.

$\frac{4 \cdot 2 - 1}{2}$

Step 6.5

Simplify the numerator.

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Step 6.5.1

Multiply $4$ by $2$ .

$\frac{8 - 1}{2}$

Step 6.5.2

Subtract $1$ from $8$ .

$\frac{7}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{7}{2}$

Decimal Form:

$3.5$