Question

f(x) = \

$$\begin{cases} \\sin(x \\cdot \\pi) & \\text{for } -8 < x < 0 \\\\ \\dfrac{x}{5} & \\text{for } 0 \\leq x \\leq 10 \\end{cases}$$

find $\lim\limits_{x \to -5} f(x)$.
choose 1 answer:
a $-1$
b $0$
c $1$
d the limit doesnt exist.

Explanation:

Step1: Determine the applicable function

Since \( -8 < -5 < 0 \), we use the function \( f(x)=\sin(x\cdot\pi) \) for this limit.

Step2: Substitute \( x = -5 \) into the function

We need to find \( \lim_{x
ightarrow -5}\sin(x\cdot\pi) \). Substitute \( x=-5 \) into \( \sin(x\pi) \), we get \( \sin((-5)\pi) \).

Step3: Simplify the sine function

We know that \( \sin(n\pi)=0 \) for any integer \( n \). Here, \( n = -5 \), so \( \sin(-5\pi)=0 \).

Answer:

B. 0