multiple choice 2 points\nquestion topic(s)/section(s): 1.11 defining continuity at a point\nclick the timer…

multiple choice 2 points\nquestion topic(s)/section(s): 1.11 defining continuity at a point\nclick the timer to pace yourself. some questions do not require the full 2 minutes, and others will require a little more than 2 minutes.\nh(x)=\\begin{cases}ae^{2x}+3&\\text{for }x < 0\\5 - cosx&\\text{for }x\\geq0\\end{cases}\nlet h be the function defined above, where a is a constant. if h is continuous at x = 0, what is the value of a?\n0\n2\n5\n1\nclear my selection

multiple choice 2 points\nquestion topic(s)/section(s): 1.11 defining continuity at a point\nclick the timer to pace yourself. some questions do not require the full 2 minutes, and others will require a little more than 2 minutes.\nh(x)=\\begin{cases}ae^{2x}+3&\\text{for }x < 0\\5 - cosx&\\text{for }x\\geq0\\end{cases}\nlet h be the function defined above, where a is a constant. if h is continuous at x = 0, what is the value of a?\n0\n2\n5\n1\nclear my selection

Answer

Explanation:

Step1: Recall continuity condition

For a function to be continuous at (x = 0), (\lim_{x\rightarrow0^{-}}h(x)=\lim_{x\rightarrow0^{+}}h(x)=h(0)).

Step2: Calculate left - hand limit

(\lim_{x\rightarrow0^{-}}h(x)=\lim_{x\rightarrow0^{-}}(ae^{2x}+3)=a\cdot e^{0}+3=a + 3).

Step3: Calculate right - hand limit and function value at (x = 0)

(\lim_{x\rightarrow0^{+}}h(x)=h(0)=5-\cos(0)=5 - 1=4).

Step4: Equate left - hand and right - hand limits

Set (a + 3=4), then (a=4 - 3=1).

Answer:

D. 1