Calculus Examples

$\frac{lim_{x \to 0} {(x - t)}^{2} - x^{2}}{t}$

Step 1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{lim_{x \to 0} {(x - t)}^{2} - lim_{x \to 0} x^{2}}{t}$

Step 2

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

$\frac{{(lim_{x \to 0} x - t)}^{2} - lim_{x \to 0} x^{2}}{t}$

Step 3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{{(lim_{x \to 0} x - lim_{x \to 0} t)}^{2} - lim_{x \to 0} x^{2}}{t}$

Step 4

Evaluate the limit of $t$ which is constant as $x$ approaches $0$ .

$\frac{{(lim_{x \to 0} x - t)}^{2} - lim_{x \to 0} x^{2}}{t}$

Step 5

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

$\frac{{(lim_{x \to 0} x - t)}^{2} - {(lim_{x \to 0} x)}^{2}}{t}$

Step 6

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

Tap for more steps...

Step 6.1

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{{(0 - t)}^{2} - {(lim_{x \to 0} x)}^{2}}{t}$

Step 6.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{{(0 - t)}^{2} - 0^{2}}{t}$

Step 7

Simplify the answer.

Tap for more steps...

Step 7.1

Simplify the numerator.

Tap for more steps...

Step 7.1.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 0 - t$ and $b = 0$ .

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

Step 7.1.2

Simplify.

Tap for more steps...

Step 7.1.2.1

Add $0 - t$ and $0$ .

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

Step 7.1.2.2

Subtract $t$ from $0$ .

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

Step 7.1.2.3

Add $0 - t$ and $0$ .

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

Step 7.1.2.4

Subtract $t$ from $0$ .

$\frac{- t \cdot - 1 t}{t}$

Step 7.1.2.5

Combine exponents.

Tap for more steps...

Step 7.1.2.5.1

Factor out negative.

$\frac{- (t \cdot - 1 t)}{t}$

Step 7.1.2.5.2

Raise $t$ to the power of $1$ .

$\frac{- - (t^{1} t)}{t}$

Step 7.1.2.5.3

Raise $t$ to the power of $1$ .

$\frac{- - (t^{1} t^{1})}{t}$

Step 7.1.2.5.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 7.1.2.5.5

Add $1$ and $1$ .

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

Step 7.1.2.5.6

Multiply $- 1$ by $- 1$ .

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

Step 7.1.2.5.7

Multiply $t^{2}$ by $1$ .

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

Step 7.2

Cancel the common factor of $t^{2}$ and $t$ .

Tap for more steps...

Step 7.2.1

Factor $t$ out of $t^{2}$ .

$\frac{t \cdot t}{t}$

Step 7.2.2

Cancel the common factors.

Tap for more steps...

Step 7.2.2.1

Raise $t$ to the power of $1$ .

$\frac{t \cdot t}{t^{1}}$

Step 7.2.2.2

Factor $t$ out of $t^{1}$ .

$\frac{t \cdot t}{t \cdot 1}$

Step 7.2.2.3

Cancel the common factor.

$\frac{t \cdot t}{t \cdot 1}$

Step 7.2.2.4

Rewrite the expression.

$\frac{t}{1}$

Step 7.2.2.5

Divide $t$ by $1$ .

$t$