Determine the limit algebraically, if possible.
-$$\lim _ { x \rightarrow 0 } x ^ { 2 } \cot x ^ { 2 }$$
A) 2
B) Does not exist
C) 0
D) 1

Question

Determine the limit algebraically, if possible.
-$$\lim _ { x \rightarrow  0 } x ^ { 2 } \cot x ^ { 2 }$$
A) 2
B) Does not exist
C) 0
D) 1

Quizplus · Accepted Answer

To solve this limit, we can use the trigonometric identity $\cot(x) = \frac{1}{\tan(x)}$, and the fact that $\lim_{x \to 0} \frac{\sin(x)}{x} = 1$. Rewriting the given limit in terms of sine and cosine gives us $\lim _ { x \rightarrow 0 } x ^ { 2 } \frac{\cos(x^2)}{\sin(x^2)}$. We can then multiply the numerator and the denominator by $\frac{1}{x^2}$ to get $\lim _ { x \rightarrow 0 } \frac{\cos(x^2)}{\frac{\sin(x^2)}{x^2}}$. Since $\lim_{x \to 0} \cos(x^2) = \cos(0) = 1$ and $\lim_{x \to 0} \frac{\sin(x^2)}{x^2} = 1$ (by substituting $x^2$ for $x$ in the limit $\lim_{x \to 0} \frac{\sin(x)}{x} = 1$), the overall limit is $1$.

Determine the Limit Algebraically, If Possible lim⁡x→0x2cot⁡x2\lim _ { x \rightarrow 0 } x ^ { 2 } \cot x ^ { 2 }x→0lim​x2cotx2

Determine the Limit Algebraically, If Possible $\lim _ { x \rightarrow 0 } x ^ { 2 } \cot x ^ { 2 }$