Calculus Examples

$lim_{t \to - 2} \frac{t^{4} - 2}{2 t^{2} - 3 t + 2}$

Step 1

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

$\frac{lim_{t \to - 2} t^{4} - 2}{lim_{t \to - 2} 2 t^{2} - 3 t + 2}$

Step 2

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

$\frac{lim_{t \to - 2} t^{4} - lim_{t \to - 2} 2}{lim_{t \to - 2} 2 t^{2} - 3 t + 2}$

Step 3

Move the exponent $4$ from $t^{4}$ outside the limit using the Limits Power Rule.

$\frac{{(lim_{t \to - 2} t)}^{4} - lim_{t \to - 2} 2}{lim_{t \to - 2} 2 t^{2} - 3 t + 2}$

Step 4

Evaluate the limit of $2$ which is constant as $t$ approaches $- 2$ .

$\frac{{(lim_{t \to - 2} t)}^{4} - 1 \cdot 2}{lim_{t \to - 2} 2 t^{2} - 3 t + 2}$

Step 5

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

$\frac{{(lim_{t \to - 2} t)}^{4} - 1 \cdot 2}{lim_{t \to - 2} 2 t^{2} - lim_{t \to - 2} 3 t + lim_{t \to - 2} 2}$

Step 6

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

$\frac{{(lim_{t \to - 2} t)}^{4} - 1 \cdot 2}{2 lim_{t \to - 2} t^{2} - lim_{t \to - 2} 3 t + lim_{t \to - 2} 2}$

Step 7

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

$\frac{{(lim_{t \to - 2} t)}^{4} - 1 \cdot 2}{2 {(lim_{t \to - 2} t)}^{2} - lim_{t \to - 2} 3 t + lim_{t \to - 2} 2}$

Step 8

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

$\frac{{(lim_{t \to - 2} t)}^{4} - 1 \cdot 2}{2 {(lim_{t \to - 2} t)}^{2} - 3 lim_{t \to - 2} t + lim_{t \to - 2} 2}$

Step 9

Evaluate the limit of $2$ which is constant as $t$ approaches $- 2$ .

$\frac{{(lim_{t \to - 2} t)}^{4} - 1 \cdot 2}{2 {(lim_{t \to - 2} t)}^{2} - 3 lim_{t \to - 2} t + 2}$

Step 10

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

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Step 10.1

Evaluate the limit of $t$ by plugging in $- 2$ for $t$ .

$\frac{{(- 2)}^{4} - 1 \cdot 2}{2 {(lim_{t \to - 2} t)}^{2} - 3 lim_{t \to - 2} t + 2}$

Step 10.2

Evaluate the limit of $t$ by plugging in $- 2$ for $t$ .

$\frac{{(- 2)}^{4} - 1 \cdot 2}{2 {(- 2)}^{2} - 3 lim_{t \to - 2} t + 2}$

Step 10.3

Evaluate the limit of $t$ by plugging in $- 2$ for $t$ .

$\frac{{(- 2)}^{4} - 1 \cdot 2}{2 {(- 2)}^{2} - 3 \cdot - 2 + 2}$

Step 11

Simplify the answer.

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Step 11.1

Simplify the numerator.

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Step 11.1.1

Raise $- 2$ to the power of $4$ .

$\frac{16 - 1 \cdot 2}{2 {(- 2)}^{2} - 3 \cdot - 2 + 2}$

Step 11.1.2

Multiply $- 1$ by $2$ .

$\frac{16 - 2}{2 {(- 2)}^{2} - 3 \cdot - 2 + 2}$

Step 11.1.3

Subtract $2$ from $16$ .

$\frac{14}{2 {(- 2)}^{2} - 3 \cdot - 2 + 2}$

Step 11.2

Simplify the denominator.

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Step 11.2.1

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

$\frac{14}{2 \cdot 4 - 3 \cdot - 2 + 2}$

Step 11.2.2

Multiply $2$ by $4$ .

$\frac{14}{8 - 3 \cdot - 2 + 2}$

Step 11.2.3

Multiply $- 3$ by $- 2$ .

$\frac{14}{8 + 6 + 2}$

Step 11.2.4

Add $8$ and $6$ .

$\frac{14}{14 + 2}$

Step 11.2.5

Add $14$ and $2$ .

$\frac{14}{16}$

Step 11.3

Cancel the common factor of $14$ and $16$ .

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Step 11.3.1

Factor $2$ out of $14$ .

$\frac{2 (7)}{16}$

Step 11.3.2

Cancel the common factors.

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Step 11.3.2.1

Factor $2$ out of $16$ .

$\frac{2 \cdot 7}{2 \cdot 8}$

Step 11.3.2.2

Cancel the common factor.

$\frac{2 \cdot 7}{2 \cdot 8}$

Step 11.3.2.3

Rewrite the expression.

$\frac{7}{8}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{7}{8}$

Decimal Form:

$0.875$