Find the limit and determine if the function is continuous at the point being approached.
-$$\lim _ { x \rightarrow 4 } \sin \left( x \sin ^ { 2 } x + x \cos ^ { 2 } x + 3 \right)$$
A) does not exist; no
B) sin -1; yes
C) sin 7; no
D) sin 7; yes

Question

Quizplus · Accepted Answer

The limit as $x$ approaches 4 can be simplified using the trigonometric identity $\sin^2x + \cos^2x = 1$. Thus, the expression inside the sine function simplifies to $x(\sin^2x + \cos^2x) + 3 = x(1) + 3 = x + 3$. Substituting $x = 4$ gives $4 + 3 = 7$, so the limit is $\sin(7)$. Since the sine function is continuous and the expression inside the sine function is a polynomial (which is also continuous), the overall function is continuous at the point $x = 4$.

Find the Limit and Determine If the Function Is Continuous lim⁡x→4sin⁡(xsin⁡2x+xcos⁡2x+3)\lim _ { x \rightarrow 4 } \sin \left( x \sin ^ { 2 } x + x \cos ^ { 2 } x + 3 \right)x→4lim​sin(xsin2x+xcos2x+3)

Find the Limit and Determine If the Function Is Continuous $\lim _ { x \rightarrow 4 } \sin \left( x \sin ^ { 2 } x + x \cos ^ { 2 } x + 3 \right)$