Calculus Examples

$lim_{x \to 3} \frac{x^{n} - 3}{x^{m} - 3}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $3$ .

$\frac{lim_{x \to 3} x^{n} - 3}{lim_{x \to 3} x^{m} - 3}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $3$ .

$\frac{lim_{x \to 3} x^{n} - lim_{x \to 3} 3}{lim_{x \to 3} x^{m} - 3}$

Step 3

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

$\frac{{(lim_{x \to 3} x)}^{n} - lim_{x \to 3} 3}{lim_{x \to 3} x^{m} - 3}$

Step 4

Evaluate the limit of $3$ which is constant as $x$ approaches $3$ .

$\frac{{(lim_{x \to 3} x)}^{n} - 1 \cdot 3}{lim_{x \to 3} x^{m} - 3}$

Step 5

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $3$ .

$\frac{{(lim_{x \to 3} x)}^{n} - 1 \cdot 3}{lim_{x \to 3} x^{m} - lim_{x \to 3} 3}$

Step 6

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

$\frac{{(lim_{x \to 3} x)}^{n} - 1 \cdot 3}{{(lim_{x \to 3} x)}^{m} - lim_{x \to 3} 3}$

Step 7

Evaluate the limit of $3$ which is constant as $x$ approaches $3$ .

$\frac{{(lim_{x \to 3} x)}^{n} - 1 \cdot 3}{{(lim_{x \to 3} x)}^{m} - 1 \cdot 3}$

Step 8

Evaluate the limits by plugging in $3$ for all occurrences of $x$ .

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Step 8.1

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

$\frac{3^{n} - 1 \cdot 3}{{(lim_{x \to 3} x)}^{m} - 1 \cdot 3}$

Step 8.2

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

$\frac{3^{n} - 1 \cdot 3}{3^{m} - 1 \cdot 3}$

Step 9

Simplify the answer.

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Step 9.1

Multiply $- 1$ by $3$ .

$\frac{3^{n} - 3}{3^{m} - 1 \cdot 3}$

Step 9.2

Multiply $- 1$ by $3$ .

$\frac{3^{n} - 3}{3^{m} - 3}$