Calculus Examples

$lim_{x \to x^{+}} \frac{3}{x^{3}} - \frac{1}{x}$

Step 1

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

$lim_{x \to x^{+}} \frac{3}{x^{3}} - lim_{x \to x^{+}} \frac{1}{x}$

Step 2

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

$3 lim_{x \to x^{+}} \frac{1}{x^{3}} - lim_{x \to x^{+}} \frac{1}{x}$

Step 3

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

$3 \frac{lim_{x \to x^{+}} 1}{lim_{x \to x^{+}} x^{3}} - lim_{x \to x^{+}} \frac{1}{x}$

Step 4

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

$3 \frac{1}{lim_{x \to x^{+}} x^{3}} - lim_{x \to x^{+}} \frac{1}{x}$

Step 5

Move the exponent $3$ from $x^{3}$ outside the limit using the Limits Power Rule.

$3 \frac{1}{{(lim_{x \to x^{+}} x)}^{3}} - lim_{x \to x^{+}} \frac{1}{x}$

Step 6

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

$3 \frac{1}{{(lim_{x \to x^{+}} x)}^{3}} - \frac{lim_{x \to x^{+}} 1}{lim_{x \to x^{+}} x}$

Step 7

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

$3 \frac{1}{{(lim_{x \to x^{+}} x)}^{3}} - \frac{1}{lim_{x \to x^{+}} x}$

Step 8

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

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Step 8.1

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

$3 \frac{1}{x^{3}} - \frac{1}{lim_{x \to x^{+}} x}$

Step 8.2

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

$3 \frac{1}{x^{3}} - \frac{1}{x}$

Step 9

Combine $3$ and $\frac{1}{x^{3}}$ .

$\frac{3}{x^{3}} - \frac{1}{x}$