Calculus Examples

$lim_{x \to 0} \frac{e^{x^{2}} - \cos (x)}{x^{2}}$

Step 1

Apply L'Hospital's rule.

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Step 1.1

Evaluate the limit of the numerator and the limit of the denominator.

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Step 1.1.1

Take the limit of the numerator and the limit of the denominator.

$\frac{lim_{x \to 0} e^{x^{2}} - \cos (x)}{lim_{x \to 0} x^{2}}$

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

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

$\frac{lim_{x \to 0} e^{x^{2}} - lim_{x \to 0} \cos (x)}{lim_{x \to 0} x^{2}}$

Step 1.1.2.2

Move the limit into the exponent.

$\frac{e^{lim_{x \to 0} x^{2}} - lim_{x \to 0} \cos (x)}{lim_{x \to 0} x^{2}}$

Step 1.1.2.3

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

$\frac{e^{{(lim_{x \to 0} x)}^{2}} - lim_{x \to 0} \cos (x)}{lim_{x \to 0} x^{2}}$

Step 1.1.2.4

Move the limit inside the trig function because cosine is continuous.

$\frac{e^{{(lim_{x \to 0} x)}^{2}} - \cos (lim_{x \to 0} x)}{lim_{x \to 0} x^{2}}$

Step 1.1.2.5

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

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Step 1.1.2.5.1

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

$\frac{e^{0^{2}} - \cos (lim_{x \to 0} x)}{lim_{x \to 0} x^{2}}$

Step 1.1.2.5.2

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

$\frac{e^{0^{2}} - \cos (0)}{lim_{x \to 0} x^{2}}$

Step 1.1.2.6

Simplify the answer.

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Step 1.1.2.6.1

Simplify each term.

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Step 1.1.2.6.1.1

Raising $0$ to any positive power yields $0$ .

$\frac{e^{0} - \cos (0)}{lim_{x \to 0} x^{2}}$

Step 1.1.2.6.1.2

Anything raised to $0$ is $1$ .

$\frac{1 - \cos (0)}{lim_{x \to 0} x^{2}}$

Step 1.1.2.6.1.3

The exact value of $\cos (0)$ is $1$ .

$\frac{1 - 1 \cdot 1}{lim_{x \to 0} x^{2}}$

Step 1.1.2.6.1.4

Multiply $- 1$ by $1$ .

$\frac{1 - 1}{lim_{x \to 0} x^{2}}$

Step 1.1.2.6.2

Subtract $1$ from $1$ .

$\frac{0}{lim_{x \to 0} x^{2}}$

Step 1.1.3

Evaluate the limit of the denominator.

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Step 1.1.3.1

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

$\frac{0}{{(lim_{x \to 0} x)}^{2}}$

Step 1.1.3.2

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

$\frac{0}{0^{2}}$

Step 1.1.3.3

Raising $0$ to any positive power yields $0$ .

$\frac{0}{0}$

Step 1.1.3.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 1.1.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 1.2

Since $\frac{0}{0}$ is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.

$lim_{x \to 0} \frac{e^{x^{2}} - \cos (x)}{x^{2}} = lim_{x \to 0} \frac{\frac{d}{d x} [e^{x^{2}} - \cos (x)]}{\frac{d}{d x} [x^{2}]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to 0} \frac{\frac{d}{d x} [e^{x^{2}} - \cos (x)]}{\frac{d}{d x} [x^{2}]}$

Step 1.3.2

By the Sum Rule, the derivative of $e^{x^{2}} - \cos (x)$ with respect to $x$ is $\frac{d}{d x} [e^{x^{2}}] + \frac{d}{d x} [- \cos (x)]$ .

$lim_{x \to 0} \frac{\frac{d}{d x} [e^{x^{2}}] + \frac{d}{d x} [- \cos (x)]}{\frac{d}{d x} [x^{2}]}$

Step 1.3.3

Evaluate $\frac{d}{d x} [e^{x^{2}}]$ .

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Step 1.3.3.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = x^{2}$ .

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Step 1.3.3.1.1

To apply the Chain Rule, set $u$ as $x^{2}$ .

$lim_{x \to 0} \frac{\frac{d}{d u} [e^{u}] \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- \cos (x)]}{\frac{d}{d x} [x^{2}]}$

Step 1.3.3.1.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $e$ .

$lim_{x \to 0} \frac{e^{u} \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- \cos (x)]}{\frac{d}{d x} [x^{2}]}$

Step 1.3.3.1.3

Replace all occurrences of $u$ with $x^{2}$ .

$lim_{x \to 0} \frac{e^{x^{2}} \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- \cos (x)]}{\frac{d}{d x} [x^{2}]}$

Step 1.3.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$lim_{x \to 0} \frac{e^{x^{2}} (2 x) + \frac{d}{d x} [- \cos (x)]}{\frac{d}{d x} [x^{2}]}$

Step 1.3.4

Evaluate $\frac{d}{d x} [- \cos (x)]$ .

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Step 1.3.4.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- \cos (x)$ with respect to $x$ is $- \frac{d}{d x} [\cos (x)]$ .

$lim_{x \to 0} \frac{e^{x^{2}} (2 x) - \frac{d}{d x} [\cos (x)]}{\frac{d}{d x} [x^{2}]}$

Step 1.3.4.2

The derivative of $\cos (x)$ with respect to $x$ is $- \sin (x)$ .

$lim_{x \to 0} \frac{e^{x^{2}} (2 x) - - \sin (x)}{\frac{d}{d x} [x^{2}]}$

Step 1.3.4.3

Multiply $- 1$ by $- 1$ .

$lim_{x \to 0} \frac{e^{x^{2}} (2 x) + 1 \sin (x)}{\frac{d}{d x} [x^{2}]}$

Step 1.3.4.4

Multiply $\sin (x)$ by $1$ .

$lim_{x \to 0} \frac{e^{x^{2}} (2 x) + \sin (x)}{\frac{d}{d x} [x^{2}]}$

Step 1.3.5

Simplify.

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Step 1.3.5.1

Reorder terms.

$lim_{x \to 0} \frac{2 e^{x^{2}} x + \sin (x)}{\frac{d}{d x} [x^{2}]}$

Step 1.3.5.2

Reorder factors in $2 e^{x^{2}} x + \sin (x)$ .

$lim_{x \to 0} \frac{2 x e^{x^{2}} + \sin (x)}{\frac{d}{d x} [x^{2}]}$

Step 1.3.6

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$lim_{x \to 0} \frac{2 x e^{x^{2}} + \sin (x)}{2 x}$

Step 2

Move the term $\frac{1}{2}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{2} lim_{x \to 0} \frac{2 x e^{x^{2}} + \sin (x)}{x}$

Step 3

Apply L'Hospital's rule.

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Step 3.1

Evaluate the limit of the numerator and the limit of the denominator.

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Step 3.1.1

Take the limit of the numerator and the limit of the denominator.

$\frac{1}{2} \cdot \frac{lim_{x \to 0} 2 x e^{x^{2}} + \sin (x)}{lim_{x \to 0} x}$

Step 3.1.2

Evaluate the limit of the numerator.

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Step 3.1.2.1

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

$\frac{1}{2} \cdot \frac{lim_{x \to 0} 2 x e^{x^{2}} + lim_{x \to 0} \sin (x)}{lim_{x \to 0} x}$

Step 3.1.2.2

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

$\frac{1}{2} \cdot \frac{2 lim_{x \to 0} x e^{x^{2}} + lim_{x \to 0} \sin (x)}{lim_{x \to 0} x}$

Step 3.1.2.3

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

$\frac{1}{2} \cdot \frac{2 lim_{x \to 0} x \cdot lim_{x \to 0} e^{x^{2}} + lim_{x \to 0} \sin (x)}{lim_{x \to 0} x}$

Step 3.1.2.4

Move the limit into the exponent.

$\frac{1}{2} \cdot \frac{2 lim_{x \to 0} x \cdot e^{lim_{x \to 0} x^{2}} + lim_{x \to 0} \sin (x)}{lim_{x \to 0} x}$

Step 3.1.2.5

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

$\frac{1}{2} \cdot \frac{2 lim_{x \to 0} x \cdot e^{{(lim_{x \to 0} x)}^{2}} + lim_{x \to 0} \sin (x)}{lim_{x \to 0} x}$

Step 3.1.2.6

Move the limit inside the trig function because sine is continuous.

$\frac{1}{2} \cdot \frac{2 lim_{x \to 0} x \cdot e^{{(lim_{x \to 0} x)}^{2}} + \sin (lim_{x \to 0} x)}{lim_{x \to 0} x}$

Step 3.1.2.7

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

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Step 3.1.2.7.1

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

$\frac{1}{2} \cdot \frac{2 \cdot 0 \cdot e^{{(lim_{x \to 0} x)}^{2}} + \sin (lim_{x \to 0} x)}{lim_{x \to 0} x}$

Step 3.1.2.7.2

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

$\frac{1}{2} \cdot \frac{2 \cdot 0 \cdot e^{0^{2}} + \sin (lim_{x \to 0} x)}{lim_{x \to 0} x}$

Step 3.1.2.7.3

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

$\frac{1}{2} \cdot \frac{2 \cdot 0 \cdot e^{0^{2}} + \sin (0)}{lim_{x \to 0} x}$

Step 3.1.2.8

Simplify the answer.

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Step 3.1.2.8.1

Simplify each term.

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Step 3.1.2.8.1.1

Multiply $2$ by $0$ .

$\frac{1}{2} \cdot \frac{0 \cdot e^{0^{2}} + \sin (0)}{lim_{x \to 0} x}$

Step 3.1.2.8.1.2

Raising $0$ to any positive power yields $0$ .

$\frac{1}{2} \cdot \frac{0 \cdot e^{0} + \sin (0)}{lim_{x \to 0} x}$

Step 3.1.2.8.1.3

Anything raised to $0$ is $1$ .

$\frac{1}{2} \cdot \frac{0 \cdot 1 + \sin (0)}{lim_{x \to 0} x}$

Step 3.1.2.8.1.4

Multiply $0$ by $1$ .

$\frac{1}{2} \cdot \frac{0 + \sin (0)}{lim_{x \to 0} x}$

Step 3.1.2.8.1.5

The exact value of $\sin (0)$ is $0$ .

$\frac{1}{2} \cdot \frac{0 + 0}{lim_{x \to 0} x}$

Step 3.1.2.8.2

Add $0$ and $0$ .

$\frac{1}{2} \cdot \frac{0}{lim_{x \to 0} x}$

Step 3.1.3

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

$\frac{1}{2} \cdot \frac{0}{0}$

Step 3.1.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{1}{2} \cdot \frac{0}{0}$

Step 3.2

Since $\frac{0}{0}$ is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.

$lim_{x \to 0} \frac{2 x e^{x^{2}} + \sin (x)}{x} = lim_{x \to 0} \frac{\frac{d}{d x} [2 x e^{x^{2}} + \sin (x)]}{\frac{d}{d x} [x]}$

Step 3.3

Find the derivative of the numerator and denominator.

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Step 3.3.1

Differentiate the numerator and denominator.

$\frac{1}{2} lim_{x \to 0} \frac{\frac{d}{d x} [2 x e^{x^{2}} + \sin (x)]}{\frac{d}{d x} [x]}$

Step 3.3.2

By the Sum Rule, the derivative of $2 x e^{x^{2}} + \sin (x)$ with respect to $x$ is $\frac{d}{d x} [2 x e^{x^{2}}] + \frac{d}{d x} [\sin (x)]$ .

$\frac{1}{2} lim_{x \to 0} \frac{\frac{d}{d x} [2 x e^{x^{2}}] + \frac{d}{d x} [\sin (x)]}{\frac{d}{d x} [x]}$

Step 3.3.3

Evaluate $\frac{d}{d x} [2 x e^{x^{2}}]$ .

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Step 3.3.3.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x e^{x^{2}}$ with respect to $x$ is $2 \frac{d}{d x} [x e^{x^{2}}]$ .

$\frac{1}{2} lim_{x \to 0} \frac{2 \frac{d}{d x} [x e^{x^{2}}] + \frac{d}{d x} [\sin (x)]}{\frac{d}{d x} [x]}$

Step 3.3.3.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = e^{x^{2}}$ .

$\frac{1}{2} lim_{x \to 0} \frac{2 (x \frac{d}{d x} [e^{x^{2}}] + e^{x^{2}} \frac{d}{d x} [x]) + \frac{d}{d x} [\sin (x)]}{\frac{d}{d x} [x]}$

Step 3.3.3.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = x^{2}$ .

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Step 3.3.3.3.1

To apply the Chain Rule, set $u$ as $x^{2}$ .

$\frac{1}{2} lim_{x \to 0} \frac{2 (x \frac{d}{d u} [e^{u}] \frac{d}{d x} [x^{2}] + e^{x^{2}} \frac{d}{d x} [x]) + \frac{d}{d x} [\sin (x)]}{\frac{d}{d x} [x]}$

Step 3.3.3.3.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $e$ .

$\frac{1}{2} lim_{x \to 0} \frac{2 (x e^{u} \frac{d}{d x} [x^{2}] + e^{x^{2}} \frac{d}{d x} [x]) + \frac{d}{d x} [\sin (x)]}{\frac{d}{d x} [x]}$

Step 3.3.3.3.3

Replace all occurrences of $u$ with $x^{2}$ .

$\frac{1}{2} lim_{x \to 0} \frac{2 (x e^{x^{2}} \frac{d}{d x} [x^{2}] + e^{x^{2}} \frac{d}{d x} [x]) + \frac{d}{d x} [\sin (x)]}{\frac{d}{d x} [x]}$

Step 3.3.3.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$\frac{1}{2} lim_{x \to 0} \frac{2 (x e^{x^{2}} \cdot 2 x + e^{x^{2}} \frac{d}{d x} [x]) + \frac{d}{d x} [\sin (x)]}{\frac{d}{d x} [x]}$

Step 3.3.3.5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$\frac{1}{2} lim_{x \to 0} \frac{2 (x e^{x^{2}} \cdot 2 x + e^{x^{2}} \cdot 1) + \frac{d}{d x} [\sin (x)]}{\frac{d}{d x} [x]}$

Step 3.3.3.6

Raise $x$ to the power of $1$ .

$\frac{1}{2} lim_{x \to 0} \frac{2 (x^{1} x e^{x^{2}} \cdot 2 + e^{x^{2}} \cdot 1) + \frac{d}{d x} [\sin (x)]}{\frac{d}{d x} [x]}$

Step 3.3.3.7

Raise $x$ to the power of $1$ .

$\frac{1}{2} lim_{x \to 0} \frac{2 (x^{1} x^{1} e^{x^{2}} \cdot 2 + e^{x^{2}} \cdot 1) + \frac{d}{d x} [\sin (x)]}{\frac{d}{d x} [x]}$

Step 3.3.3.8

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{1}{2} lim_{x \to 0} \frac{2 (x^{1 + 1} e^{x^{2}} \cdot 2 + e^{x^{2}} \cdot 1) + \frac{d}{d x} [\sin (x)]}{\frac{d}{d x} [x]}$

Step 3.3.3.9

Add $1$ and $1$ .

$\frac{1}{2} lim_{x \to 0} \frac{2 (x^{2} e^{x^{2}} \cdot 2 + e^{x^{2}} \cdot 1) + \frac{d}{d x} [\sin (x)]}{\frac{d}{d x} [x]}$

Step 3.3.3.10

Move $2$ to the left of $e^{x^{2}}$ .

$\frac{1}{2} lim_{x \to 0} \frac{2 (x^{2} \cdot 2 \cdot e^{x^{2}} + e^{x^{2}} \cdot 1) + \frac{d}{d x} [\sin (x)]}{\frac{d}{d x} [x]}$

Step 3.3.3.11

Multiply $e^{x^{2}}$ by $1$ .

$\frac{1}{2} lim_{x \to 0} \frac{2 (x^{2} \cdot 2 e^{x^{2}} + e^{x^{2}}) + \frac{d}{d x} [\sin (x)]}{\frac{d}{d x} [x]}$

Step 3.3.4

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

$\frac{1}{2} lim_{x \to 0} \frac{2 (x^{2} \cdot 2 e^{x^{2}} + e^{x^{2}}) + \cos (x)}{\frac{d}{d x} [x]}$

Step 3.3.5

Simplify.

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Step 3.3.5.1

Apply the distributive property.

$\frac{1}{2} lim_{x \to 0} \frac{2 x^{2} \cdot 2 e^{x^{2}} + 2 e^{x^{2}} + \cos (x)}{\frac{d}{d x} [x]}$

Step 3.3.5.2

Multiply $2$ by $2$ .

$\frac{1}{2} lim_{x \to 0} \frac{4 x^{2} e^{x^{2}} + 2 e^{x^{2}} + \cos (x)}{\frac{d}{d x} [x]}$

Step 3.3.5.3

Reorder terms.

$\frac{1}{2} lim_{x \to 0} \frac{4 e^{x^{2}} x^{2} + 2 e^{x^{2}} + \cos (x)}{\frac{d}{d x} [x]}$

Step 3.3.5.4

Reorder factors in $4 e^{x^{2}} x^{2} + 2 e^{x^{2}} + \cos (x)$ .

$\frac{1}{2} lim_{x \to 0} \frac{4 x^{2} e^{x^{2}} + 2 e^{x^{2}} + \cos (x)}{\frac{d}{d x} [x]}$

Step 3.3.6

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$\frac{1}{2} lim_{x \to 0} \frac{4 x^{2} e^{x^{2}} + 2 e^{x^{2}} + \cos (x)}{1}$

Step 3.4

Divide $4 x^{2} e^{x^{2}} + 2 e^{x^{2}} + \cos (x)$ by $1$ .

$\frac{1}{2} lim_{x \to 0} 4 x^{2} e^{x^{2}} + 2 e^{x^{2}} + \cos (x)$

Step 4

Evaluate the limit.

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Step 4.1

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

$\frac{1}{2} (lim_{x \to 0} 4 x^{2} e^{x^{2}} + lim_{x \to 0} 2 e^{x^{2}} + lim_{x \to 0} \cos (x))$

Step 4.2

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

$\frac{1}{2} (4 lim_{x \to 0} x^{2} e^{x^{2}} + lim_{x \to 0} 2 e^{x^{2}} + lim_{x \to 0} \cos (x))$

Step 4.3

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

$\frac{1}{2} (4 lim_{x \to 0} x^{2} \cdot lim_{x \to 0} e^{x^{2}} + lim_{x \to 0} 2 e^{x^{2}} + lim_{x \to 0} \cos (x))$

Step 4.4

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

$\frac{1}{2} (4 {(lim_{x \to 0} x)}^{2} \cdot lim_{x \to 0} e^{x^{2}} + lim_{x \to 0} 2 e^{x^{2}} + lim_{x \to 0} \cos (x))$

Step 4.5

Move the limit into the exponent.

$\frac{1}{2} (4 {(lim_{x \to 0} x)}^{2} \cdot e^{lim_{x \to 0} x^{2}} + lim_{x \to 0} 2 e^{x^{2}} + lim_{x \to 0} \cos (x))$

Step 4.6

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

$\frac{1}{2} (4 {(lim_{x \to 0} x)}^{2} \cdot e^{{(lim_{x \to 0} x)}^{2}} + lim_{x \to 0} 2 e^{x^{2}} + lim_{x \to 0} \cos (x))$

Step 4.7

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

$\frac{1}{2} (4 {(lim_{x \to 0} x)}^{2} \cdot e^{{(lim_{x \to 0} x)}^{2}} + 2 lim_{x \to 0} e^{x^{2}} + lim_{x \to 0} \cos (x))$

Step 4.8

Move the limit into the exponent.

$\frac{1}{2} (4 {(lim_{x \to 0} x)}^{2} \cdot e^{{(lim_{x \to 0} x)}^{2}} + 2 e^{lim_{x \to 0} x^{2}} + lim_{x \to 0} \cos (x))$

Step 4.9

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

$\frac{1}{2} (4 {(lim_{x \to 0} x)}^{2} \cdot e^{{(lim_{x \to 0} x)}^{2}} + 2 e^{{(lim_{x \to 0} x)}^{2}} + lim_{x \to 0} \cos (x))$

Step 4.10

Move the limit inside the trig function because cosine is continuous.

$\frac{1}{2} (4 {(lim_{x \to 0} x)}^{2} \cdot e^{{(lim_{x \to 0} x)}^{2}} + 2 e^{{(lim_{x \to 0} x)}^{2}} + \cos (lim_{x \to 0} x))$

Step 5

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

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Step 5.1

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

$\frac{1}{2} (4 \cdot 0^{2} \cdot e^{{(lim_{x \to 0} x)}^{2}} + 2 e^{{(lim_{x \to 0} x)}^{2}} + \cos (lim_{x \to 0} x))$

Step 5.2

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

$\frac{1}{2} (4 \cdot 0^{2} \cdot e^{0^{2}} + 2 e^{{(lim_{x \to 0} x)}^{2}} + \cos (lim_{x \to 0} x))$

Step 5.3

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

$\frac{1}{2} (4 \cdot 0^{2} \cdot e^{0^{2}} + 2 e^{0^{2}} + \cos (lim_{x \to 0} x))$

Step 5.4

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

$\frac{1}{2} (4 \cdot 0^{2} \cdot e^{0^{2}} + 2 e^{0^{2}} + \cos (0))$

Step 6

Simplify the answer.

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Step 6.1

Simplify each term.

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Step 6.1.1

Raising $0$ to any positive power yields $0$ .

$\frac{1}{2} (4 \cdot 0 \cdot e^{0^{2}} + 2 e^{0^{2}} + \cos (0))$

Step 6.1.2

Multiply $4$ by $0$ .

$\frac{1}{2} (0 \cdot e^{0^{2}} + 2 e^{0^{2}} + \cos (0))$

Step 6.1.3

Raising $0$ to any positive power yields $0$ .

$\frac{1}{2} (0 \cdot e^{0} + 2 e^{0^{2}} + \cos (0))$

Step 6.1.4

Anything raised to $0$ is $1$ .

$\frac{1}{2} (0 \cdot 1 + 2 e^{0^{2}} + \cos (0))$

Step 6.1.5

Multiply $0$ by $1$ .

$\frac{1}{2} (0 + 2 e^{0^{2}} + \cos (0))$

Step 6.1.6

Raising $0$ to any positive power yields $0$ .

$\frac{1}{2} (0 + 2 e^{0} + \cos (0))$

Step 6.1.7

Anything raised to $0$ is $1$ .

$\frac{1}{2} (0 + 2 \cdot 1 + \cos (0))$

Step 6.1.8

Multiply $2$ by $1$ .

$\frac{1}{2} (0 + 2 + \cos (0))$

Step 6.1.9

The exact value of $\cos (0)$ is $1$ .

$\frac{1}{2} (0 + 2 + 1)$

Step 6.2

Add $0$ and $2$ .

$\frac{1}{2} (2 + 1)$

Step 6.3

Add $2$ and $1$ .

$\frac{1}{2} \cdot 3$

Step 6.4

Combine $\frac{1}{2}$ and $3$ .

$\frac{3}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{3}{2}$

Decimal Form:

$1.5$