Calculus Examples

$lim_{x \to 3.99} \frac{x - 4}{x^{2} - 3 x - 4}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $3.99$ .

$\frac{lim_{x \to 3.99} x - 4}{lim_{x \to 3.99} x^{2} - 3 x - 4}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $3.99$ .

$\frac{lim_{x \to 3.99} x - lim_{x \to 3.99} 4}{lim_{x \to 3.99} x^{2} - 3 x - 4}$

Step 3

Evaluate the limit of $4$ which is constant as $x$ approaches $3.99$ .

$\frac{lim_{x \to 3.99} x - 1 \cdot 4}{lim_{x \to 3.99} x^{2} - 3 x - 4}$

Step 4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $3.99$ .

$\frac{lim_{x \to 3.99} x - 1 \cdot 4}{lim_{x \to 3.99} x^{2} - lim_{x \to 3.99} 3 x - lim_{x \to 3.99} 4}$

Step 5

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

$\frac{lim_{x \to 3.99} x - 1 \cdot 4}{{(lim_{x \to 3.99} x)}^{2} - lim_{x \to 3.99} 3 x - lim_{x \to 3.99} 4}$

Step 6

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

$\frac{lim_{x \to 3.99} x - 1 \cdot 4}{{(lim_{x \to 3.99} x)}^{2} - 3 lim_{x \to 3.99} x - lim_{x \to 3.99} 4}$

Step 7

Evaluate the limit of $4$ which is constant as $x$ approaches $3.99$ .

$\frac{lim_{x \to 3.99} x - 1 \cdot 4}{{(lim_{x \to 3.99} x)}^{2} - 3 lim_{x \to 3.99} x - 1 \cdot 4}$

Step 8

Evaluate the limits by plugging in $3.99$ for all occurrences of $x$ .

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Step 8.1

Evaluate the limit of $x$ by plugging in $3.99$ for $x$ .

$\frac{3.99 - 1 \cdot 4}{{(lim_{x \to 3.99} x)}^{2} - 3 lim_{x \to 3.99} x - 1 \cdot 4}$

Step 8.2

Evaluate the limit of $x$ by plugging in $3.99$ for $x$ .

$\frac{3.99 - 1 \cdot 4}{{3.99}^{2} - 3 lim_{x \to 3.99} x - 1 \cdot 4}$

Step 8.3

Evaluate the limit of $x$ by plugging in $3.99$ for $x$ .

$\frac{3.99 - 1 \cdot 4}{{3.99}^{2} - 3 \cdot 3.99 - 1 \cdot 4}$

Step 9

Simplify the answer.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Multiply $- 1$ by $4$ .

$\frac{3.99 - 4}{{3.99}^{2} - 3 \cdot 3.99 - 1 \cdot 4}$

Step 9.1.2

Subtract $4$ from $3.99$ .

$\frac{- 0.01}{{3.99}^{2} - 3 \cdot 3.99 - 1 \cdot 4}$

Step 9.2

Simplify the denominator.

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Step 9.2.1

Raise $3.99$ to the power of $2$ .

$\frac{- 0.01}{15.9201 - 3 \cdot 3.99 - 1 \cdot 4}$

Step 9.2.2

Multiply $- 3$ by $3.99$ .

$\frac{- 0.01}{15.9201 - 11.97 - 1 \cdot 4}$

Step 9.2.3

Multiply $- 1$ by $4$ .

$\frac{- 0.01}{15.9201 - 11.97 - 4}$

Step 9.2.4

Subtract $11.97$ from $15.9201$ .

$\frac{- 0.01}{3.9501 - 4}$

Step 9.2.5

Subtract $4$ from $3.9501$ .

$\frac{- 0.01}{- 0.0499}$

Step 9.3

Divide $- 0.01$ by $- 0.0499$ .

$0.2004008$