Calculus/Limits/Exercises
<
Calculus
<
Limits
lim
x
→
∞
[
cos
(
1
x
)
]
x
3
log
(
1
+
1
x
)
{\displaystyle \lim _{x\to \infty }\left[\cos {\left({\frac {1}{x}}\right)}\right]^{x^{3}\log {(1+{{1} \over {x}})}}}
lim
x
→
∞
1
x
{\displaystyle \lim _{x\to \infty }{\frac {1}{x}}}
in
R
+
{\displaystyle \mathbb {R} ^{+}}
Answer: 0.
lim
n
→
∞
(
n
+
1
)
α
−
n
α
n
α
−
1
{\displaystyle \lim _{n\to \infty }{\frac {{\left(n+1\right)}^{\alpha }-n^{\alpha }}{n^{\alpha -1}}}}
Answer:
α
{\displaystyle \alpha }
lim
x
→
∞
x
log
(
1
+
e
x
)
−
x
x
{\displaystyle \lim _{x\to \infty }{\sqrt {x}}\log {\left(1+e^{x}\right)}-x{\sqrt {x}}}
Answer: 0
lim
x
→
∞
x
log
(
1
+
e
x
)
−
x
x
−
1
{\displaystyle \lim _{x\to \infty }{\sqrt {x}}\log {\left(1+e^{x}\right)}-x{\sqrt {x-1}}}
Answer:
−
∞
{\displaystyle -\infty }
lim
x
→
∞
x
+
sin
x
x
(
2
+
sin
x
)
x
x
!
{\displaystyle \lim _{x\to \infty }{\frac {{\sqrt[{x}]{x+\sin {x}}}\left(2+\sin {x}\right)^{x}}{x!}}}
lim
x
→
∞
x
3
+
x
−
x
x
x
+
6
sin
x
{\displaystyle \lim _{x\to \infty }{\frac {{\sqrt {x^{3}+x}}-x{\sqrt {x}}}{x+6\sin {x}}}}
![{\displaystyle \lim _{x\to \infty }\left[\cos {\left({\frac {1}{x}}\right)}\right]^{x^{3}\log {(1+{{1} \over {x}})}}}](../../4282576e446549a6137a51be86a3deb784e7644c.svg)
in 
![{\displaystyle \lim _{x\to \infty }{\frac {{\sqrt[{x}]{x+\sin {x}}}\left(2+\sin {x}\right)^{x}}{x!}}}](../../39fcc23b89f51e14b54ded2696a0608d4ee1c1aa.svg)
