Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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Step 4.1

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

$\frac{1^{2} + lim_{t \to 1} t - 1 \cdot 2}{t^{2} - 1}$

Step 4.2

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

$\frac{1^{2} + 1 - 1 \cdot 2}{t^{2} - 1}$

Step 5

Simplify the answer.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

One to any power is one.

$\frac{1 + 1 - 1 \cdot 2}{t^{2} - 1}$

Step 5.1.2

Multiply $- 1$ by $2$ .

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

Step 5.1.3

Add $1$ and $1$ .

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

Step 5.1.4

Subtract $2$ from $2$ .

$\frac{0}{t^{2} - 1}$

Step 5.2

Simplify the denominator.

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Step 5.2.1

Rewrite $1$ as $1^{2}$ .

$\frac{0}{t^{2} - 1^{2}}$

Step 5.2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = t$ and $b = 1$ .

$\frac{0}{(t + 1) (t - 1)}$

Step 5.3

Divide $0$ by $(t + 1) (t - 1)$ .

$0$