Evaluate the following limit ( IE do not

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

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

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

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

Evaluate the limit of the numerator.

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

Evaluate the limit of the denominator.

${\displaystyle \frac{0}{0}}$

The expression contains a division by 0${\displaystyle 0}$. The expression is undefined.

Undefined

Since ${\displaystyle \frac{0}{0}}$ 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.

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

Find the derivative of the numerator and denominator.

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

Cancel the common factor of ${\displaystyle 2x-4}$ and ${\displaystyle 2x-10}$.

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

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

Cancel the common factors.

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

Split the limit using the Limits Quotient Rule on the limit as x${\displaystyle x}$ approaches ${\displaystyle 4}$.

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

Split the limit using the Product of Limits Rule on the limit as ${\displaystyle x}$ approaches ${\displaystyle 4}$.

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

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

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

Split the limit using the Sum of Limits Rule on the limit as ${\displaystyle x}$ approaches ${\displaystyle 4}$.

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

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

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

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

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

Split the limit using the Sum of Limits Rule on the limit as ${\displaystyle x}$ approaches ${\displaystyle 4}$.

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

Evaluate the limit of ${\displaystyle 2}$ which is constant as x${\displaystyle x}$ approaches ${\displaystyle 4}$.

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

Split the limit using the Product of Limits Rule on the limit as ${\displaystyle x}$ approaches ${\displaystyle 4}$.

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

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

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

Split the limit using the Sum of Limits Rule on the limit as x${\displaystyle x}$ approaches 4${\displaystyle 4}$.

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

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

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

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

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

Evaluate the limit of ${\displaystyle 24}$ which is constant as x${\displaystyle x}$ approaches ${\displaystyle 4}$.

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

Split the limit using the Sum of Limits Rule on the limit as ${\displaystyle x}$ approaches ${\displaystyle 4}$.

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

Evaluate the limit of ${\displaystyle 5}$ which is constant as ${\displaystyle x}$ approaches ${\displaystyle 4}$.

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

Evaluate the limits by plugging in ${\displaystyle 4}$ for all occurrences of ${\displaystyle x}$.

Evaluate the limit of ${\displaystyle x}$ by plugging in ${\displaystyle 4}$ for ${\displaystyle x}$.

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

Evaluate the limit of ${\displaystyle x}$ by plugging in ${\displaystyle 4}$ for ${\displaystyle x}$.

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

Evaluate the limit of ${\displaystyle x}$ by plugging in ${\displaystyle 4}$ for ${\displaystyle x}$.

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

Evaluate the limit of ${\displaystyle x}$ by plugging in ${\displaystyle 4}$ for ${\displaystyle x}$.

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

Evaluate the limit of ${\displaystyle x}$ by plugging in ${\displaystyle 4}$ for ${\displaystyle x}$.

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

Evaluate the limit of ${\displaystyle x}$ by plugging in ${\displaystyle 4}$ for ${\displaystyle x}$.

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

Simplify the numerator.

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

Simplify the denominator.

${\displaystyle \frac{2}{-1}}$

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

${\displaystyle -1\cdot 2}$

Multiply ${\displaystyle -1}$ by ${\displaystyle 2}$.

$-2$