What is the limit as x approaches 5-of (sin((x-7)(x-5)))/(x-5) ?

The limit as x approaches 5-of (sin((x-7)(x-5)))/(x-5) is -2

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limit as x approaches 5-of (sin((x-7)(x-5)))/(x-5)

x \to 5 - lim (\frac{sin ( ( x - 7 ) ( x - 5 ) )}{x - 5})

- 2

Solution steps

x \to 5 - lim (\frac{sin ( ( x - 7 ) ( x - 5 ) )}{x - 5})

Factor out

x

from

x - 7 : x (1 - \frac{7}{x})

= x \to 5 - lim (\frac{sin ( x ( 1 - \frac{7}{x} ) ( x - 5 ) )}{x - 5})

Expand

x (1 - \frac{7}{x}) (x - 5) : x^{2} + 35 - 12 x

= x \to 5 - lim (\frac{sin ( x ^{2} + 35 - 12 x )}{x - 5})

Apply L'Hopital's Rule

: x \to 5 - lim (\frac{cos ( x ^{2} + 35 - 12 x ) ( 2 x - 12 )}{1})

= x \to 5 - lim (\frac{cos ( x ^{2} + 35 - 12 x ) ( 2 x - 12 )}{1})

Plug in the value

x = 5

= \frac{cos ( 5 ^{2} + 35 - 12 \cdot 5 ) ( 2 \cdot 5 - 12 )}{1}

Simplify

\frac{cos ( 5 ^{2} + 35 - 12 \cdot 5 ) ( 2 \cdot 5 - 12 )}{1} : - 2

= - 2

x \to 1 lim (1 + x^{\frac{1}{1 - x}})

h \to 0 lim (\frac{\frac{2}{8 + h} - 8}{h})

x \to - 2 lim (- x^{2} + 3 x + 4)

x \to 0 lim (a^{m x})

h \to 0 lim (\frac{1 - cos ( h )}{h})

What is the limit as x approaches 5-of (sin((x-7)(x-5)))/(x-5) ?
The limit as x approaches 5-of (sin((x-7)(x-5)))/(x-5) is -2