Calculus Examples

$lim_{x \to 7} 49 - \frac{x^{2}}{7 + x}$

Step 1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $7$ .

$lim_{x \to 7} 49 - lim_{x \to 7} \frac{x^{2}}{7 + x}$

Step 2

Evaluate the limit of $49$ which is constant as $x$ approaches $7$ .

$49 - lim_{x \to 7} \frac{x^{2}}{7 + x}$

Step 3

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $7$ .

$49 - \frac{lim_{x \to 7} x^{2}}{lim_{x \to 7} 7 + x}$

Step 4

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

$49 - \frac{{(lim_{x \to 7} x)}^{2}}{lim_{x \to 7} 7 + x}$

Step 5

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $7$ .

$49 - \frac{{(lim_{x \to 7} x)}^{2}}{lim_{x \to 7} 7 + lim_{x \to 7} x}$

Step 6

Evaluate the limit of $7$ which is constant as $x$ approaches $7$ .

$49 - \frac{{(lim_{x \to 7} x)}^{2}}{7 + lim_{x \to 7} x}$

Step 7

Evaluate the limits by plugging in $7$ for all occurrences of $x$ .

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Step 7.1

Evaluate the limit of $x$ by plugging in $7$ for $x$ .

$49 - \frac{7^{2}}{7 + lim_{x \to 7} x}$

Step 7.2

Evaluate the limit of $x$ by plugging in $7$ for $x$ .

$49 - \frac{7^{2}}{7 + 7}$

Step 8

Simplify the answer.

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Step 8.1

Simplify each term.

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Step 8.1.1

Raise $7$ to the power of $2$ .

$49 - \frac{49}{7 + 7}$

Step 8.1.2

Add $7$ and $7$ .

$49 - \frac{49}{14}$

Step 8.1.3

Cancel the common factor of $49$ and $14$ .

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Step 8.1.3.1

Factor $7$ out of $49$ .

$49 - \frac{7 (7)}{14}$

Step 8.1.3.2

Cancel the common factors.

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Step 8.1.3.2.1

Factor $7$ out of $14$ .

$49 - \frac{7 \cdot 7}{7 \cdot 2}$

Step 8.1.3.2.2

Cancel the common factor.

$49 - \frac{7 \cdot 7}{7 \cdot 2}$

Step 8.1.3.2.3

Rewrite the expression.

$49 - \frac{7}{2}$

Step 8.2

To write $49$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$49 \cdot \frac{2}{2} - \frac{7}{2}$

Step 8.3

Combine $49$ and $\frac{2}{2}$ .

$\frac{49 \cdot 2}{2} - \frac{7}{2}$

Step 8.4

Combine the numerators over the common denominator.

$\frac{49 \cdot 2 - 7}{2}$

Step 8.5

Simplify the numerator.

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Step 8.5.1

Multiply $49$ by $2$ .

$\frac{98 - 7}{2}$

Step 8.5.2

Subtract $7$ from $98$ .

$\frac{91}{2}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{91}{2}$

Decimal Form:

$45.5$