Calculus Examples

$lim_{x \to 0} \frac{1 + x}{4 x - 1}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $0$ .

$\frac{lim_{x \to 0} 1 + x}{lim_{x \to 0} 4 x - 1}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{lim_{x \to 0} 1 + lim_{x \to 0} x}{lim_{x \to 0} 4 x - 1}$

Step 3

Evaluate the limit of $1$ which is constant as $x$ approaches $0$ .

$\frac{1 + lim_{x \to 0} x}{lim_{x \to 0} 4 x - 1}$

Step 4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{1 + lim_{x \to 0} x}{lim_{x \to 0} 4 x - lim_{x \to 0} 1}$

Step 5

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$\frac{1 + lim_{x \to 0} x}{4 lim_{x \to 0} x - lim_{x \to 0} 1}$

Step 6

Evaluate the limit of $1$ which is constant as $x$ approaches $0$ .

$\frac{1 + lim_{x \to 0} x}{4 lim_{x \to 0} x - 1 \cdot 1}$

Step 7

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

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Step 7.1

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{1 + 0}{4 lim_{x \to 0} x - 1 \cdot 1}$

Step 7.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{1 + 0}{4 \cdot 0 - 1 \cdot 1}$

Step 8

Simplify the answer.

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Step 8.1

Add $1$ and $0$ .

$\frac{1}{4 \cdot 0 - 1 \cdot 1}$

Step 8.2

Simplify the denominator.

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Step 8.2.1

Multiply $4$ by $0$ .

$\frac{1}{0 - 1 \cdot 1}$

Step 8.2.2

Multiply $- 1$ by $1$ .

$\frac{1}{0 - 1}$

Step 8.2.3

Subtract $1$ from $0$ .

$\frac{1}{- 1}$

Step 8.3

Divide $1$ by $- 1$ .

$- 1$