1. A function f (x) is defined as follows:
x² +1
when x > 0
when x = 0 , Find the value of lim f (x).
when x < 0
so
x+1
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Answered by
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Answer:
hope brainliest
Step-by-step explanation:
Answer
h→0
lim
f(0+h)=1+sinh=1=f(0−h)
Hence, f(x) is continuous at x=0.
Now, f
′
(0
+
)=
h→0
lim
h
1+sinh−1
=
h→0
lim
h
sinh
=1
f
′
(0
−
)=
h→0
lim
h
1−1
=0
Hence, f(x) is not differentiable at x=0.
h→0
lim
f(
2
π
−h)=
h→0
lim
1+sin(
2
π
−h)=2
h→0
lim
f(
2
π
+h)=
h→0
lim
2+(h)
2
=2
Hence, f(x) is continuous at x=
2
π
Now, f
′
(
+
)=
h→0
lim
h
2+h
2
−2
=0
f
′
(
−
)=
h→0
lim
−h
1+sin(
2
π
−h)−2
=
h→0
lim
−h
cos(h)−1
=
h→0
lim
−1
−sin(h)
=0
Hence, f(x) is also differentiable at x=
2
π
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