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AP® Calculus AB-BC

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Continuity by Finding $a$ with Piecewise Trig and L'Hospital's

APCALC-N$3NYW

Find the $a$ that makes $f(x)$ continuous for all real numbers.

$$f(x)=\begin{Bmatrix} \frac { sin(2x) }{ x } +a & x<0 \\\ 3cos(x) & 0\le x \end{Bmatrix}$$

A

$a=0$

B

$a=1$

C

$a=3$

D

There is no value of $a$ that can make $f(x)$ continuous since you get: $\lim _{ x\rightarrow { 0}^{ - } } \cfrac { sin(2x) }{ x } +a=\cfrac { sin(0) }{0 } +a$.