Calculus Examples

$lim_{x \to 5} \frac{3 x^{2} - 7}{9 - x}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $5$ .

$\frac{lim_{x \to 5} 3 x^{2} - 7}{lim_{x \to 5} 9 - x}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $5$ .

$\frac{lim_{x \to 5} 3 x^{2} - lim_{x \to 5} 7}{lim_{x \to 5} 9 - x}$

Step 3

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$\frac{3 lim_{x \to 5} x^{2} - lim_{x \to 5} 7}{lim_{x \to 5} 9 - x}$

Step 4

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

$\frac{3 {(lim_{x \to 5} x)}^{2} - lim_{x \to 5} 7}{lim_{x \to 5} 9 - x}$

Step 5

Evaluate the limit of $7$ which is constant as $x$ approaches $5$ .

$\frac{3 {(lim_{x \to 5} x)}^{2} - 1 \cdot 7}{lim_{x \to 5} 9 - x}$

Step 6

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $5$ .

$\frac{3 {(lim_{x \to 5} x)}^{2} - 1 \cdot 7}{lim_{x \to 5} 9 - lim_{x \to 5} x}$

Step 7

Evaluate the limit of $9$ which is constant as $x$ approaches $5$ .

$\frac{3 {(lim_{x \to 5} x)}^{2} - 1 \cdot 7}{9 - lim_{x \to 5} x}$

Step 8

Evaluate the limits by plugging in $5$ for all occurrences of $x$ .

Tap for more steps...

Step 8.1

Evaluate the limit of $x$ by plugging in $5$ for $x$ .

$\frac{3 \cdot 5^{2} - 1 \cdot 7}{9 - lim_{x \to 5} x}$

Step 8.2

Evaluate the limit of $x$ by plugging in $5$ for $x$ .

$\frac{3 \cdot 5^{2} - 1 \cdot 7}{9 - 5}$

Step 9

Simplify the answer.

Tap for more steps...

Step 9.1

Simplify the numerator.

Tap for more steps...

Step 9.1.1

Raise $5$ to the power of $2$ .

$\frac{3 \cdot 25 - 1 \cdot 7}{9 - 5}$

Step 9.1.2

Multiply $3$ by $25$ .

$\frac{75 - 1 \cdot 7}{9 - 5}$

Step 9.1.3

Multiply $- 1$ by $7$ .

$\frac{75 - 7}{9 - 5}$

Step 9.1.4

Subtract $7$ from $75$ .

$\frac{68}{9 - 5}$

Step 9.2

Subtract $5$ from $9$ .

$\frac{68}{4}$

Step 9.3

Divide $68$ by $4$ .

$17$