Calculus Examples

$lim_{t \to - 3} \frac{6 + 4 t}{t^{2} + 1}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $t$ approaches $- 3$ .

$\frac{lim_{t \to - 3} 6 + 4 t}{lim_{t \to - 3} t^{2} + 1}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $t$ approaches $- 3$ .

$\frac{lim_{t \to - 3} 6 + lim_{t \to - 3} 4 t}{lim_{t \to - 3} t^{2} + 1}$

Step 3

Evaluate the limit of $6$ which is constant as $t$ approaches $- 3$ .

$\frac{6 + lim_{t \to - 3} 4 t}{lim_{t \to - 3} t^{2} + 1}$

Step 4

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

$\frac{6 + 4 lim_{t \to - 3} t}{lim_{t \to - 3} t^{2} + 1}$

Step 5

Split the limit using the Sum of Limits Rule on the limit as $t$ approaches $- 3$ .

$\frac{6 + 4 lim_{t \to - 3} t}{lim_{t \to - 3} t^{2} + lim_{t \to - 3} 1}$

Step 6

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

$\frac{6 + 4 lim_{t \to - 3} t}{{(lim_{t \to - 3} t)}^{2} + lim_{t \to - 3} 1}$

Step 7

Evaluate the limit of $1$ which is constant as $t$ approaches $- 3$ .

$\frac{6 + 4 lim_{t \to - 3} t}{{(lim_{t \to - 3} t)}^{2} + 1}$

Step 8

Evaluate the limits by plugging in $- 3$ for all occurrences of $t$ .

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Step 8.1

Evaluate the limit of $t$ by plugging in $- 3$ for $t$ .

$\frac{6 + 4 \cdot - 3}{{(lim_{t \to - 3} t)}^{2} + 1}$

Step 8.2

Evaluate the limit of $t$ by plugging in $- 3$ for $t$ .

$\frac{6 + 4 \cdot - 3}{{(- 3)}^{2} + 1}$

Step 9

Simplify the answer.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Multiply $4$ by $- 3$ .

$\frac{6 - 12}{{(- 3)}^{2} + 1}$

Step 9.1.2

Subtract $12$ from $6$ .

$\frac{- 6}{{(- 3)}^{2} + 1}$

Step 9.2

Simplify the denominator.

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Step 9.2.1

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

$\frac{- 6}{9 + 1}$

Step 9.2.2

Add $9$ and $1$ .

$\frac{- 6}{10}$

Step 9.3

Cancel the common factor of $- 6$ and $10$ .

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Step 9.3.1

Factor $2$ out of $- 6$ .

$\frac{2 (- 3)}{10}$

Step 9.3.2

Cancel the common factors.

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Step 9.3.2.1

Factor $2$ out of $10$ .

$\frac{2 \cdot - 3}{2 \cdot 5}$

Step 9.3.2.2

Cancel the common factor.

$\frac{2 \cdot - 3}{2 \cdot 5}$

Step 9.3.2.3

Rewrite the expression.

$\frac{- 3}{5}$

Step 9.4

Move the negative in front of the fraction.

$- \frac{3}{5}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$- \frac{3}{5}$

Decimal Form:

$- 0.6$