Calculus Examples

$lim_{x \to \frac{1}{4}} 2 x^{2} - 7$

Step 1

Evaluate the limit.

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Step 1.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $\frac{1}{4}$ .

$lim_{x \to \frac{1}{4}} 2 x^{2} - lim_{x \to \frac{1}{4}} 7$

Step 1.2

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

$2 lim_{x \to \frac{1}{4}} x^{2} - lim_{x \to \frac{1}{4}} 7$

Step 1.3

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

$2 {(lim_{x \to \frac{1}{4}} x)}^{2} - lim_{x \to \frac{1}{4}} 7$

Step 1.4

Evaluate the limit of $7$ which is constant as $x$ approaches $\frac{1}{4}$ .

$2 {(lim_{x \to \frac{1}{4}} x)}^{2} - 1 \cdot 7$

Step 2

Evaluate the limit of $x$ by plugging in $\frac{1}{4}$ for $x$ .

$2 {(\frac{1}{4})}^{2} - 1 \cdot 7$

Step 3

Simplify the answer.

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Step 3.1

Simplify each term.

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Step 3.1.1

Apply the product rule to $\frac{1}{4}$ .

$2 \frac{1^{2}}{4^{2}} - 1 \cdot 7$

Step 3.1.2

One to any power is one.

$2 \frac{1}{4^{2}} - 1 \cdot 7$

Step 3.1.3

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

$2 (\frac{1}{16}) - 1 \cdot 7$

Step 3.1.4

Cancel the common factor of $2$ .

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Step 3.1.4.1

Factor $2$ out of $16$ .

$2 \frac{1}{2 (8)} - 1 \cdot 7$

Step 3.1.4.2

Cancel the common factor.

$2 \frac{1}{2 \cdot 8} - 1 \cdot 7$

Step 3.1.4.3

Rewrite the expression.

$\frac{1}{8} - 1 \cdot 7$

Step 3.1.5

Multiply $- 1$ by $7$ .

$\frac{1}{8} - 7$

Step 3.2

To write $- 7$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$\frac{1}{8} - 7 \cdot \frac{8}{8}$

Step 3.3

Combine $- 7$ and $\frac{8}{8}$ .

$\frac{1}{8} + \frac{- 7 \cdot 8}{8}$

Step 3.4

Combine the numerators over the common denominator.

$\frac{1 - 7 \cdot 8}{8}$

Step 3.5

Simplify the numerator.

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Step 3.5.1

Multiply $- 7$ by $8$ .

$\frac{1 - 56}{8}$

Step 3.5.2

Subtract $56$ from $1$ .

$\frac{- 55}{8}$

Step 3.6

Move the negative in front of the fraction.

$- \frac{55}{8}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$- \frac{55}{8}$

Decimal Form:

$- 6.875$