Calculus Examples

$lim_{x \to 1} \frac{- e^{x}}{x + 4 x^{2}}$

Step 1

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

$\frac{lim_{x \to 1} - e^{x}}{lim_{x \to 1} x + 4 x^{2}}$

Step 2

Move the term $- 1$ outside of the limit because it is constant with respect to $x$ .

$\frac{- lim_{x \to 1} e^{x}}{lim_{x \to 1} x + 4 x^{2}}$

Step 3

Move the limit into the exponent.

$\frac{- e^{lim_{x \to 1} x}}{lim_{x \to 1} x + 4 x^{2}}$

Step 4

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

$\frac{- e^{lim_{x \to 1} x}}{lim_{x \to 1} x + lim_{x \to 1} 4 x^{2}}$

Step 5

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

$\frac{- e^{lim_{x \to 1} x}}{lim_{x \to 1} x + 4 lim_{x \to 1} x^{2}}$

Step 6

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

$\frac{- e^{lim_{x \to 1} x}}{lim_{x \to 1} x + 4 {(lim_{x \to 1} x)}^{2}}$

Step 7

Evaluate the limits by plugging in $1$ for all occurrences of $x$ .

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Step 7.1

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

$\frac{- e^{1}}{lim_{x \to 1} x + 4 {(lim_{x \to 1} x)}^{2}}$

Step 7.2

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

$\frac{- e^{1}}{1 + 4 {(lim_{x \to 1} x)}^{2}}$

Step 7.3

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

$\frac{- e^{1}}{1 + 4 \cdot 1^{2}}$

Step 8

Simplify the answer.

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Step 8.1

Simplify.

$\frac{- e}{1 + 4 \cdot 1^{2}}$

Step 8.2

Simplify the denominator.

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Step 8.2.1

One to any power is one.

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

Step 8.2.2

Multiply $4$ by $1$ .

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

Step 8.2.3

Add $1$ and $4$ .

$\frac{- e}{5}$

Step 8.3

Move the negative in front of the fraction.

$- \frac{e}{5}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$- \frac{e}{5}$

Decimal Form:

$- 0.54365636 \dots$