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$Let f(x)=(x)/(5sin(x+1)). Select the correct description of the one-sided limits of f at x=-1. Choose 1 answer: (A) {:[lim_(x rarr-1^(+))f(x)=+oo" and "],[lim_(x rarr-1^(-))f(x)=+oo]:} (B) {:[lim_(x rarr-1^(+))f(x)=+oo" and "],[lim_(x rarr-1^(-))f(x)=-oo]:} (c) {:[lim_(x rarr-1^(+))f(x)=-oo" and "],[lim_(x rarr-1^(-))f(x)=+oo]:} (D) {:[lim_(x rarr-1^(+))f(x)=-oo" and "],[lim_(x rarr-1^(-))f(x)=-oo]:}$

Let $f(x)=\frac{x}{5 \sin (x+1)}$ . $\newline$ Select the correct description of the one-sided limits of $f$ at $x=-1$ . $\newline$ Choose $1$ answer: $\newline$ (A) $\newline$ $\begin{array}{l} \lim _{x \rightarrow-1^{+}} f(x)=+\infty \text { and } \\ \lim _{x \rightarrow-1^{-}} f(x)=+\infty \end{array}$ $\newline$ (B) $\newline$ $\begin{array}{l} \lim _{x \rightarrow-1^{+}} f(x)=+\infty \text { and } \\ \lim _{x \rightarrow-1^{-}} f(x)=-\infty \end{array}$ $\newline$ (C) $\newline$ $\begin{array}{l} \lim _{x \rightarrow-1^{+}} f(x)=-\infty \text { and } \\ \lim _{x \rightarrow-1^{-}} f(x)=+\infty \end{array}$ $\newline$ (D) $\newline$ $\begin{array}{l} \lim _{x \rightarrow-1^{+}} f(x)=-\infty \text { and } \\ \lim _{x \rightarrow-1^{-}} f(x)=-\infty \end{array}$

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Algebra 1

Compare linear, exponential, and quadratic growth

Full solution

Q. Let

f(x)=\frac{x}{5 \sin (x+1)}

\newline

Select the correct description of the one-sided limits of

f

x=-1

\newline

Choose

1

answer:

\newline

(A)

\newline

\begin{array}{l} \lim _{x \rightarrow-1^{+}} f(x)=+\infty \text { and } \\ \lim _{x \rightarrow-1^{-}} f(x)=+\infty \end{array}

\newline

(B)

\newline

\begin{array}{l} \lim _{x \rightarrow-1^{+}} f(x)=+\infty \text { and } \\ \lim _{x \rightarrow-1^{-}} f(x)=-\infty \end{array}

\newline

(C)

\newline

\begin{array}{l} \lim _{x \rightarrow-1^{+}} f(x)=-\infty \text { and } \\ \lim _{x \rightarrow-1^{-}} f(x)=+\infty \end{array}

\newline

(D)

\newline

\begin{array}{l} \lim _{x \rightarrow-1^{+}} f(x)=-\infty \text { and } \\ \lim _{x \rightarrow-1^{-}} f(x)=-\infty \end{array}

Analyze Function Behavior: Analyze the function near the point of interest. $\newline$ We are interested in the behavior of the function $f(x) = \frac{x}{5\sin(x+1)}$ as $x$ approaches $-1$ . To understand the behavior of the function near $x = -1$ , we need to look at the values of the sine function near $x = -1$ .
Evaluate Sine Function: Evaluate the sine function at the point of interest. $\newline$ We need to evaluate $\sin(x+1)$ as $x$ approaches $-1$ . When $x = -1$ , $x+1 = 0$ , and we know that $\sin(0) = 0$ . This means that as $x$ approaches $-1$ , the denominator of our function approaches $0$ .
Determine Sine Sign: Determine the sign of the sine function just before and just after $x = -1$ . Just before $x = -1$ (as $x$ approaches $-1$ from the left), $x+1$ is slightly negative, and since the sine function is continuous and smooth, $\sin(x+1)$ will be slightly negative. Just after $x = -1$ (as $x$ approaches $-1$ from the right), $x+1$ is slightly positive, and $\sin(x+1)$ will be slightly positive.
Calculate One-Sided Limits: Determine the one-sided limits using the sign of the sine function. $\newline$ As $x$ approaches $-1$ from the left, $\sin(x+1)$ is negative, and since the numerator $x$ is also negative, the fraction $\frac{x}{5\sin(x+1)}$ will be positive. Therefore, $\lim_{x \rightarrow -1^{-}}f(x) = +\infty$ . $\newline$ As $x$ approaches $-1$ from the right, $\sin(x+1)$ is positive, and since the numerator $x$ is negative, the fraction $\frac{x}{5\sin(x+1)}$ will be negative. Therefore, $-1$ $1$ .