Calculus Examples

$lim_{x \to 1} \frac{x^{2} - 4}{4}$

Step 1

Evaluate the limit.

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Step 1.1

Move the term $\frac{1}{4}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{4} lim_{x \to 1} x^{2} - 4$

Step 1.2

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

$\frac{1}{4} (lim_{x \to 1} x^{2} - lim_{x \to 1} 4)$

Step 1.3

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

$\frac{1}{4} ({(lim_{x \to 1} x)}^{2} - lim_{x \to 1} 4)$

Step 1.4

Evaluate the limit of $4$ which is constant as $x$ approaches $1$ .

$\frac{1}{4} ({(lim_{x \to 1} x)}^{2} - 1 \cdot 4)$

Step 2

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

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

Step 3

Simplify the answer.

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Step 3.1

Simplify each term.

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Step 3.1.1

One to any power is one.

$\frac{1}{4} (1 - 1 \cdot 4)$

Step 3.1.2

Multiply $- 1$ by $4$ .

$\frac{1}{4} (1 - 4)$

Step 3.2

Subtract $4$ from $1$ .

$\frac{1}{4} \cdot - 3$

Step 3.3

Combine $\frac{1}{4}$ and $- 3$ .

$\frac{- 3}{4}$

Step 3.4

Move the negative in front of the fraction.

$- \frac{3}{4}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$- \frac{3}{4}$

Decimal Form:

$- 0.75$