Calculus Examples

$lim_{x \to - 1} \frac{\frac{1}{4 x + 3} + 1}{4 x + 4}$

Step 1

Combine terms.

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Step 1.1

Write $1$ as a fraction with a common denominator.

$lim_{x \to - 1} \frac{\frac{1}{4 x + 3} + \frac{4 x + 3}{4 x + 3}}{4 x + 4}$

Step 1.2

Combine the numerators over the common denominator.

$lim_{x \to - 1} \frac{\frac{1 + 4 x + 3}{4 x + 3}}{4 x + 4}$

Step 1.3

Add $1$ and $3$ .

$lim_{x \to - 1} \frac{\frac{4 x + 4}{4 x + 3}}{4 x + 4}$

Step 2

Evaluate the limit.

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Step 2.1

Simplify the limit argument.

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Step 2.1.1

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to - 1} \frac{4 x + 4}{4 x + 3} \cdot \frac{1}{4 x + 4}$

Step 2.1.2

Multiply $\frac{4 x + 4}{4 x + 3}$ by $\frac{1}{4 x + 4}$ .

$lim_{x \to - 1} \frac{4 x + 4}{(4 x + 3) (4 x + 4)}$

Step 2.1.3

Cancel the common factor of $4 x + 4$ .

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Step 2.1.3.1

Cancel the common factor.

$lim_{x \to - 1} \frac{4 x + 4}{(4 x + 3) (4 x + 4)}$

Step 2.1.3.2

Rewrite the expression.

$lim_{x \to - 1} \frac{1}{4 x + 3}$

Step 2.2

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $- 1$ .

$\frac{lim_{x \to - 1} 1}{lim_{x \to - 1} 4 x + 3}$

Step 2.3

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

$\frac{1}{lim_{x \to - 1} 4 x + 3}$

Step 2.4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 1$ .

$\frac{1}{lim_{x \to - 1} 4 x + lim_{x \to - 1} 3}$

Step 2.5

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

$\frac{1}{4 lim_{x \to - 1} x + lim_{x \to - 1} 3}$

Step 2.6

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

$\frac{1}{4 lim_{x \to - 1} x + 3}$

Step 3

Evaluate the limit of $x$ by plugging in $- 1$ for $x$ .

$\frac{1}{4 \cdot - 1 + 3}$

Step 4

Simplify the answer.

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Step 4.1

Simplify the denominator.

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Step 4.1.1

Multiply $4$ by $- 1$ .

$\frac{1}{- 4 + 3}$

Step 4.1.2

Add $- 4$ and $3$ .

$\frac{1}{- 1}$

Step 4.2

Divide $1$ by $- 1$ .

$- 1$