overall accuracy: 86.8% record: 28 score: 28 evaluate lim h→0 (sin(x + h) - sin x)/h cos x 1/√(1 - x²) - cos…

overall accuracy: 86.8% record: 28 score: 28 evaluate lim h→0 (sin(x + h) - sin x)/h cos x 1/√(1 - x²) - cos x -1/√(1 - x²) high score board: overall you must have at least 100 to be on the board. # name record answer submitting is not possible on this problem. just follow the instructions above.

overall accuracy: 86.8% record: 28 score: 28 evaluate lim h→0 (sin(x + h) - sin x)/h cos x 1/√(1 - x²) - cos x -1/√(1 - x²) high score board: overall you must have at least 100 to be on the board. # name record answer submitting is not possible on this problem. just follow the instructions above.

Answer

Explanation:

Step1: Use the trigonometric identity

$\sin(A + B)=\sin A\cos B+\cos A\sin B$. So, $\sin(x + h)=\sin x\cos h+\cos x\sin h$. Then the limit becomes $\lim_{h\rightarrow0}\frac{\sin x\cos h+\cos x\sin h-\sin x}{h}$.

Step2: Rearrange the terms

$\lim_{h\rightarrow0}\frac{\sin x(\cos h - 1)+\cos x\sin h}{h}=\lim_{h\rightarrow0}\left(\sin x\frac{\cos h - 1}{h}+\cos x\frac{\sin h}{h}\right)$.

Step3: Recall known limits

We know that $\lim_{h\rightarrow0}\frac{\cos h - 1}{h}=0$ and $\lim_{h\rightarrow0}\frac{\sin h}{h}=1$.

Step4: Evaluate the limit

Substituting the known - limit values, we get $\sin x\times0+\cos x\times1=\cos x$.

Answer:

$\cos x$