Evaluate limit as x approaches 0 of (sin(x^2))/x

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

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

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

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

Evaluate the limit of the numerator.

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Take the limit of each term.

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

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

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

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

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

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

Simplify the answer.

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

$\frac{\sin (0)}{lim_{x \to 0} x}$

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

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

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

$\frac{0}{0}$

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

Undefined

$\frac{0}{0}$

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.

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

Find the derivative of the numerator and denominator.

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Differentiate the numerator and denominator.

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

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

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

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

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

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

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

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

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

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

Remove parentheses.

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

Move $2$ to the left of $\cos (x^{2})$ .

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

Multiply $2$ by $\cos (x^{2})$ .

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

Reorder the factors of $2 \cos (x^{2}) x$ .

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

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

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

Take the limit of each term.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Simplify the answer.

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Divide $2 \cdot 0 \cdot \cos (0^{2})$ by $1$ .

$2 \cdot 0 \cdot \cos (0^{2})$

Multiply $2$ by $0$ .

$0 \cdot \cos (0^{2})$

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

$0 \cdot \cos (0)$

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

$0 \cdot 1$

Multiply $0$ by $1$ .

$0$

Evaluate limit as x approaches 0 of (sin(x^2))/x

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