Calculus Examples

$lim_{x \to 3} 2 x - \frac{6}{5 x^{2 - 45}}$

Step 1

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

$lim_{x \to 3} 2 x - lim_{x \to 3} \frac{6}{5 x^{2 - 45}}$

Step 2

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

$2 lim_{x \to 3} x - lim_{x \to 3} \frac{6}{5 x^{2 - 45}}$

Step 3

Move the term $\frac{6}{5}$ outside of the limit because it is constant with respect to $x$ .

$2 lim_{x \to 3} x - \frac{6}{5} lim_{x \to 3} \frac{1}{x^{2 - 45}}$

Step 4

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

$2 lim_{x \to 3} x - \frac{6}{5} \cdot \frac{lim_{x \to 3} 1}{lim_{x \to 3} x^{2 - 45}}$

Step 5

Evaluate the limit of $1$ which is constant as $x$ approaches $3$ .

$2 lim_{x \to 3} x - \frac{6}{5} \cdot \frac{1}{lim_{x \to 3} x^{2 - 45}}$

Step 6

Move the exponent $2 - 45$ from $x^{2 - 45}$ outside the limit using the Limits Power Rule.

$2 lim_{x \to 3} x - \frac{6}{5} \cdot \frac{1}{{(lim_{x \to 3} x)}^{2 - 45}}$

Step 7

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

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Step 7.1

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

$2 \cdot 3 - \frac{6}{5} \cdot \frac{1}{{(lim_{x \to 3} x)}^{2 - 45}}$

Step 7.2

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

$2 \cdot 3 - \frac{6}{5} \cdot \frac{1}{3^{2 - 45}}$

Step 8

Simplify the answer.

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Step 8.1

Simplify each term.

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Step 8.1.1

Multiply $2$ by $3$ .

$6 - \frac{6}{5} \cdot \frac{1}{3^{2 - 45}}$

Step 8.1.2

Move $3^{2 - 45}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$6 - \frac{6}{5} \cdot 3^{- (2 - 45)}$

Step 8.1.3

Subtract $45$ from $2$ .

$6 - \frac{6}{5} \cdot 3^{- - 43}$

Step 8.1.4

Multiply $- 1$ by $- 43$ .

$6 - \frac{6}{5} \cdot 3^{43}$

Step 8.1.5

Raise $3$ to the power of $43$ .

$6 - \frac{6}{5} \cdot 328256967394537000000$

Step 8.1.6

Cancel the common factor of $5$ .

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Step 8.1.6.1

Move the leading negative in $- \frac{6}{5}$ into the numerator.

$6 + \frac{- 6}{5} \cdot 328256967394537000000$

Step 8.1.6.2

Factor $5$ out of $328256967394537000000$ .

$6 + \frac{- 6}{5} \cdot (5 (65651393478907400000))$

Step 8.1.6.3

Cancel the common factor.

$6 + \frac{- 6}{5} \cdot (5 \cdot 65651393478907400000)$

Step 8.1.6.4

Rewrite the expression.

$6 - 6 \cdot 65651393478907400000$

Step 8.1.7

Multiply $- 6$ by $65651393478907400000$ .

$6 - 393908360873444400000$

Step 8.2

Subtract $393908360873444400000$ from $6$ .

$- 393908360873444399994$

Step 9

The result can be shown in multiple forms.

Exact Form:

$- 393908360873444399994$

Decimal Form:

$- 3.9390836 \cdot 10^{20}$