2022-03-20

Evaluate the following limit ( IE do not use L'Hopital's rule).

lim𝑥→4 sin(𝑥2−4𝑥)  sin(𝑥2−10𝑥+24) .

### Answer & Explanation

alenahelenash

$\underset{x\to 4}{lim}\frac{\mathrm{sin}\left({x}^{2}-4x\right)}{\mathrm{sin}\left({x}^{2}-10x+24\right)}$

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

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

$\frac{\underset{x\to 4}{lim}\mathrm{sin}\left({x}^{2}-4x\right)}{\underset{x\to 4}{lim}\mathrm{sin}\left({x}^{2}-10x+24\right)}$

Evaluate the limit of the numerator.

$\frac{0}{\underset{x\to 4}{lim}\mathrm{sin}\left({x}^{2}-10x+24\right)}$

Evaluate the limit of the denominator.

$\frac{0}{0}$

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

Undefined

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.

$\underset{x\to 4}{lim}\frac{\mathrm{sin}\left({x}^{2}-4x\right)}{\mathrm{sin}\left({x}^{2}-10x+24\right)}=\underset{x\to 4}{lim}\frac{\frac{d}{dx}\left[\mathrm{sin}\left({x}^{2}-4x\right)\right]}{\frac{d}{dx}\left[\mathrm{sin}\left({x}^{2}-10x+24\right)\right]}$

Find the derivative of the numerator and denominator.

$\underset{x\to 4}{lim}\frac{\mathrm{cos}\left({x}^{2}-4x\right)\left(2x-4\right)}{\mathrm{cos}\left({x}^{2}-10x+24\right)\left(2x-10\right)}$

Cancel the common factor of $2x-4$ and $2x-10$.

Factor $2$ out of$\mathrm{cos}\left({x}^{2}-4x\right)\left(2x-4\right)$.

$\underset{x\to 4}{lim}\frac{2\left(\mathrm{cos}\left({x}^{2}-4x\right)\left(x-2\right)\right)}{\mathrm{cos}\left({x}^{2}-10x+24\right)\left(2x-10\right)}$

Cancel the common factors.

$\underset{x\to 4}{lim}\frac{\mathrm{cos}\left({x}^{2}-4x\right)\left(x-2\right)}{\mathrm{cos}\left({x}^{2}-10x+24\right)\left(x-5\right)}$

Split the limit using the Limits Quotient Rule on the limit as x$x$ approaches $4$.

$\frac{\underset{x\to 4}{lim}\mathrm{cos}\left({x}^{2}-4x\right)\left(x-2\right)}{\underset{x\to 4}{lim}\mathrm{cos}\left({x}^{2}-10x+24\right)\left(x-5\right)}$

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $4$.

$\frac{\underset{x\to 4}{lim}\mathrm{cos}\left({x}^{2}-4x\right)\cdot \underset{x\to 4}{lim}x-2}{\underset{x\to 4}{lim}\mathrm{cos}\left({x}^{2}-10x+24\right)\left(x-5\right)}$

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

$\frac{\mathrm{cos}\left(\underset{x\to 4}{lim}{x}^{2}-4x\right)\cdot \underset{x\to 4}{lim}x-2}{\underset{x\to 4}{lim}\mathrm{cos}\left({x}^{2}-10x+24\right)\left(x-5\right)}$

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $4$.

$\frac{\mathrm{cos}\left(\underset{x\to 4}{lim}{x}^{2}-\underset{x\to 4}{lim}4x\right)\cdot \underset{x\to 4}{lim}x-2}{\underset{x\to 4}{lim}\mathrm{cos}\left({x}^{2}-10x+24\right)\left(x-5\right)}$

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

$\frac{\mathrm{cos}\left({\left(\underset{x\to 4}{lim}x\right)}^{2}-\underset{x\to 4}{lim}4x\right)\cdot \underset{x\to 4}{lim}x-2}{\underset{x\to 4}{lim}\mathrm{cos}\left({x}^{2}-10x+24\right)\left(x-5\right)}$

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

$\frac{\mathrm{cos}\left({\left(\underset{x\to 4}{lim}x\right)}^{2}-\left(4\underset{x\to 4}{lim}x\right)\right)\cdot \underset{x\to 4}{lim}x-2}{\underset{x\to 4}{lim}\mathrm{cos}\left({x}^{2}-10x+24\right)\left(x-5\right)}$

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $4$.

$\frac{\mathrm{cos}\left({\left(\underset{x\to 4}{lim}x\right)}^{2}-1\cdot 4\underset{x\to 4}{lim}x\right)\cdot \left(\underset{x\to 4}{lim}x-\underset{x\to 4}{lim}2\right)}{\underset{x\to 4}{lim}\mathrm{cos}\left({x}^{2}-10x+24\right)\left(x-5\right)}$

Evaluate the limit of $2$ which is constant as x$x$ approaches $4$.

$\frac{\mathrm{cos}\left({\left(\underset{x\to 4}{lim}x\right)}^{2}-1\cdot 4\underset{x\to 4}{lim}x\right)\cdot \left(\underset{x\to 4}{lim}x-1\cdot 2\right)}{\underset{x\to 4}{lim}\mathrm{cos}\left({x}^{2}-10x+24\right)\left(x-5\right)}$

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $4$.

$\frac{\mathrm{cos}\left({\left(\underset{x\to 4}{lim}x\right)}^{2}-1\cdot 4\underset{x\to 4}{lim}x\right)\cdot \left(\underset{x\to 4}{lim}x-1\cdot 2\right)}{\underset{x\to 4}{lim}\mathrm{cos}\left({x}^{2}-10x+24\right)\cdot \underset{x\to 4}{lim}x-5}$

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

$\frac{\mathrm{cos}\left({\left(\underset{x\to 4}{lim}x\right)}^{2}-1\cdot 4\underset{x\to 4}{lim}x\right)\cdot \left(\underset{x\to 4}{lim}x-1\cdot 2\right)}{\mathrm{cos}\left(\underset{x\to 4}{lim}{x}^{2}-10x+24\right)\cdot \underset{x\to 4}{lim}x-5}$

Split the limit using the Sum of Limits Rule on the limit as x$x$ approaches 4$4$.

$\frac{\mathrm{cos}\left({\left(\underset{x\to 4}{lim}x\right)}^{2}-1\cdot 4\underset{x\to 4}{lim}x\right)\cdot \left(\underset{x\to 4}{lim}x-1\cdot 2\right)}{\mathrm{cos}\left(\underset{x\to 4}{lim}{x}^{2}-\underset{x\to 4}{lim}10x+\underset{x\to 4}{lim}24\right)\cdot \underset{x\to 4}{lim}x-5}$

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

$\frac{\mathrm{cos}\left({\left(\underset{x\to 4}{lim}x\right)}^{2}-1\cdot 4\underset{x\to 4}{lim}x\right)\cdot \left(\underset{x\to 4}{lim}x-1\cdot 2\right)}{\mathrm{cos}\left({\left(\underset{x\to 4}{lim}x\right)}^{2}-\underset{x\to 4}{lim}10x+\underset{x\to 4}{lim}24\right)\cdot \underset{x\to 4}{lim}x-5}$

Move the term $10$ outside of the limit because it is constant with respect to $x$.

$\frac{\mathrm{cos}\left({\left(\underset{x\to 4}{lim}x\right)}^{2}-1\cdot 4\underset{x\to 4}{lim}x\right)\cdot \left(\underset{x\to 4}{lim}x-1\cdot 2\right)}{\mathrm{cos}\left({\left(\underset{x\to 4}{lim}x\right)}^{2}-1\cdot 10\underset{x\to 4}{lim}x+\underset{x\to 4}{lim}24\right)\cdot \underset{x\to 4}{lim}x-5}$

Evaluate the limit of $24$ which is constant as x$x$ approaches $4$.

$\frac{\mathrm{cos}\left({\left(\underset{x\to 4}{lim}x\right)}^{2}-1\cdot 4\underset{x\to 4}{lim}x\right)\cdot \left(\underset{x\to 4}{lim}x-1\cdot 2\right)}{\mathrm{cos}\left({\left(\underset{x\to 4}{lim}x\right)}^{2}-1\cdot 10\underset{x\to 4}{lim}x+24\right)\cdot \underset{x\to 4}{lim}x-5}$

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $4$.

$\frac{\mathrm{cos}\left({\left(\underset{x\to 4}{lim}x\right)}^{2}-1\cdot 4\underset{x\to 4}{lim}x\right)\cdot \left(\underset{x\to 4}{lim}x-1\cdot 2\right)}{\mathrm{cos}\left({\left(\underset{x\to 4}{lim}x\right)}^{2}-1\cdot 10\underset{x\to 4}{lim}x+24\right)\cdot \left(\underset{x\to 4}{lim}x-\underset{x\to 4}{lim}5\right)}$

Evaluate the limit of $5$ which is constant as $x$ approaches $4$.

$\frac{\mathrm{cos}\left({\left(\underset{x\to 4}{lim}x\right)}^{2}-1\cdot 4\underset{x\to 4}{lim}x\right)\cdot \left(\underset{x\to 4}{lim}x-1\cdot 2\right)}{\mathrm{cos}\left({\left(\underset{x\to 4}{lim}x\right)}^{2}-1\cdot 10\underset{x\to 4}{lim}x+24\right)\cdot \left(\underset{x\to 4}{lim}x-1\cdot 5\right)}$

Evaluate the limits by plugging in $4$ for all occurrences of $x$.

Evaluate the limit of $x$ by plugging in $4$ for $x$.

$\frac{\mathrm{cos}\left({4}^{2}-1\cdot 4\underset{x\to 4}{lim}x\right)\cdot \left(\underset{x\to 4}{lim}x-1\cdot 2\right)}{\mathrm{cos}\left({\left(\underset{x\to 4}{lim}x\right)}^{2}-1\cdot 10\underset{x\to 4}{lim}x+24\right)\cdot \left(\underset{x\to 4}{lim}x-1\cdot 5\right)}$

Evaluate the limit of $x$ by plugging in $4$ for $x$.

$\frac{\mathrm{cos}\left({4}^{2}-1\cdot 4\cdot 4\right)\cdot \left(\underset{x\to 4}{lim}x-1\cdot 2\right)}{\mathrm{cos}\left({\left(\underset{x\to 4}{lim}x\right)}^{2}-1\cdot 10\underset{x\to 4}{lim}x+24\right)\cdot \left(\underset{x\to 4}{lim}x-1\cdot 5\right)}$

Evaluate the limit of $x$ by plugging in $4$ for $x$.

$\frac{\mathrm{cos}\left({4}^{2}-1\cdot 4\cdot 4\right)\cdot \left(4-1\cdot 2\right)}{\mathrm{cos}\left({\left(\underset{x\to 4}{lim}x\right)}^{2}-1\cdot 10\underset{x\to 4}{lim}x+24\right)\cdot \left(\underset{x\to 4}{lim}x-1\cdot 5\right)}$

Evaluate the limit of $x$ by plugging in $4$ for $x$.

$\frac{\mathrm{cos}\left({4}^{2}-1\cdot 4\cdot 4\right)\cdot \left(4-1\cdot 2\right)}{\mathrm{cos}\left({4}^{2}-1\cdot 10\underset{x\to 4}{lim}x+24\right)\cdot \left(\underset{x\to 4}{lim}x-1\cdot 5\right)}$

Evaluate the limit of $x$ by plugging in $4$ for $x$.

$\frac{\mathrm{cos}\left({4}^{2}-1\cdot 4\cdot 4\right)\cdot \left(4-1\cdot 2\right)}{\mathrm{cos}\left({4}^{2}-1\cdot 10\cdot 4+24\right)\cdot \left(\underset{x\to 4}{lim}x-1\cdot 5\right)}$

Evaluate the limit of $x$ by plugging in $4$ for $x$.

$\frac{\mathrm{cos}\left({4}^{2}-1\cdot 4\cdot 4\right)\cdot \left(4-1\cdot 2\right)}{\mathrm{cos}\left({4}^{2}-1\cdot 10\cdot 4+24\right)\cdot \left(4-1\cdot 5\right)}$

Simplify the numerator.

$\frac{2}{\mathrm{cos}\left({4}^{2}-1\cdot 10\cdot 4+24\right)\cdot \left(4-1\cdot 5\right)}$

Simplify the denominator.

$\frac{2}{-1}$

Move the negative one from the denominator of $\frac{2}{-1}$.

$-1\cdot 2$

Multiply $-1$ by $2$.

$-2$

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