Let $f(x)=\frac{x}{1-\cos (x-2)}$ . $\newline$ Select the correct description of the one-sided limits of $f$ at $x=2$ . $\newline$ Choose $1$ answer: $\newline$ (A) $\lim _{x \rightarrow 2^{+}} f(x)=+\infty$ and $\lim _{x \rightarrow 2^{-}} f(x)=+\infty$ $\newline$ (B) $\lim _{x \rightarrow 2^{+}} f(x)=+\infty$ and $\lim _{x \rightarrow 2^{-}} f(x)=-\infty$ $\newline$ (C) $\lim _{x \rightarrow 2^{+}} f(x)=-\infty$ and $\lim _{x \rightarrow 2^{-}} f(x)=+\infty$ $\newline$ (D) $\lim _{x \rightarrow 2^{+}} f(x)=-\infty$ and $\lim _{x \rightarrow 2^{-}} f(x)=-\infty$

View step-by-step help

Home

Math Problems

Algebra 1

Compare linear, exponential, and quadratic growth

Full solution

Q. Let

f(x)=\frac{x}{1-\cos (x-2)}

\newline

Select the correct description of the one-sided limits of

f

x=2

\newline

Choose

1

answer:

\newline

(A)

\lim _{x \rightarrow 2^{+}} f(x)=+\infty

and

\lim _{x \rightarrow 2^{-}} f(x)=+\infty

\newline

(B)

\lim _{x \rightarrow 2^{+}} f(x)=+\infty

and

\lim _{x \rightarrow 2^{-}} f(x)=-\infty

\newline

(C)

\lim _{x \rightarrow 2^{+}} f(x)=-\infty

and

\lim _{x \rightarrow 2^{-}} f(x)=+\infty

\newline

(D)

\lim _{x \rightarrow 2^{+}} f(x)=-\infty

and

\lim _{x \rightarrow 2^{-}} f(x)=-\infty

Analyze Function Behavior: Analyze the function near the point of interest. $\newline$ We are interested in the behavior of the function $f(x) = \frac{x}{1 - \cos(x - 2)}$ as $x$ approaches $2$ . Specifically, we want to know the one-sided limits as $x$ approaches $2$ from the left ( $x \to 2^{-}$ ) and from the right ( $x \to 2^{+}$ ).
Consider Denominator Behavior: Consider the behavior of the denominator as $x$ approaches $2$ . The denominator of $f(x)$ is $1 - \cos(x - 2)$ . As $x$ approaches $2$ , $x - 2$ approaches $0$ . The cosine of $0$ is $1$ , so the denominator approaches $2$ $0$ . We need to determine the sign of the denominator as $x$ approaches $2$ from the left and right to understand the behavior of $f(x)$ .
Sign of Denominator (Left): Determine the sign of the denominator as $x$ approaches $2$ from the left. $\newline$ As $x$ approaches $2$ from the left ( $x \to 2^{-}$ ), $x - 2$ is slightly negative. The cosine function is even, so $\cos(x - 2)$ is the same as $\cos(2 - x)$ . Since $2 - x$ is slightly positive when $x$ is just less than $2$ , and the cosine function is decreasing in the interval $2$ $1$ , $\cos(2 - x)$ will be slightly less than $2$ $3$ . Therefore, $2$ $4$ will be slightly positive, and the denominator will approach $2$ $5$ from the positive side. This means that $2$ $6$ will approach positive infinity.
Sign of Denominator (Right): Determine the sign of the denominator as $x$ approaches $2$ from the right. $\newline$ As $x$ approaches $2$ from the right ( $x \to 2^{(+)}$ ), $x - 2$ is slightly positive. Since the cosine function is even, $\cos(x - 2)$ will be slightly less than $1$ for the same reasons as in Step $3$ . Therefore, $1 - \cos(x - 2)$ will be slightly positive, and the denominator will approach $0$ from the positive side. This means that $2$ $0$ will approach positive infinity.
Combine Results: Combine the results from Steps $3$ and $4$ to answer the question. $\newline$ As $x$ approaches $2$ from both the left and the right, the denominator of $f(x)$ approaches $0$ from the positive side, causing $f(x)$ to approach positive infinity in both cases. Therefore, the correct description of the one-sided limits of $f$ at $x = 2$ is: $\newline$ $\lim_{x \to 2^{+}} f(x) = +\infty$ and $\lim_{x \to 2^{-}} f(x) = +\infty$ .