Chapter 2 | Limits
207
187. | x −4−2 | < 0.1, whenever | x −8| < δ In the following exercises, use the precise definition of limit to prove the given limits. 188. lim x →2 (5 x +8) =18
201. Using precise definitions of limits, prove that lim x →0 f ( x ) does not exist, given that f ( x ) is the ceiling function. ( Hint : Try any δ <1.) 202. Using precise definitions of limits, prove that lim x →0 f ( x ) does not exist: f ( x ) = ⎧ ⎩ ⎨ 1 if x is rational 0 if x is irrational . ( Hint : Think about how you can always choose a rational number 0< r < d , but | f ( r )−0 | =1.) 203. Using precise definitions of limits, determine lim x →0 f ( x ) for f ( x ) = ⎧ ⎩ ⎨ x if x is rational 0 if x is irrational into two cases, x rational and x irrational.) 204. Using the function from the previous exercise, use the precise definition of limits to show that lim x → a f ( x ) does not exist for a ≠0. For the following exercises, suppose that lim x → a f ( x ) = L and lim x → a g ( x ) = M both exist. Use the precise definition of limits to prove the following limit laws: 205. lim x → a ⎛ ⎝ f ( x )+ g ( x ) ⎞ ⎠ = L + M . ( Hint : Break
x 2 −9 x −3 =6
189. lim x →3
2 x 2 −3 x −2
190. lim x →2
x −2 =5
x 4 =0
191. lim x →0
( x 2 +2 x ) =8
192. lim x →2
In the following exercises, use the precise definition of limit to prove the given one-sided limits.
lim x →5 −
5− x =0
193.
194.
f ( x ) = −2, where f ( x ) = ⎧ ⎩
⎨ 8 x −3, if x <0 4 x −2, if x ≥0 .
lim x →0 +
⎡ ⎣ cf ( x ) ⎤
lim x
⎦ = cL for any real constant c ( Hint :
206.
→ a
f ( x ) =3, where f ( x ) = ⎧ ⎩
⎨ 5 x −2, if x <1 7 x −1, if x ≥1 .
Consider two cases: c =0 and c ≠0.)
lim x →1 −
195.
207. ⎦ = LM . ( Hint : | f ( x ) g ( x )− LM | = | f ( x ) g ( x )− f ( x ) M + f ( x ) M − LM | ≤ | f ( x ) | | g ( x )− M | + | M | | f ( x )− L | .) lim x → a ⎡ ⎣ f ( x ) g ( x ) ⎤
In the following exercises, use the precise definition of limit to prove the given infinite limits. 196. lim x →0 1 x 2 =∞
3 ( x +1) 2
lim x →−1
=∞
197.
198. lim x →2
− 1
=−∞
( x −2) 2
199. An engineer is using a machine to cut a flat square of Aerogel of area 144 cm 2 . If there is a maximum error tolerance in the area of 8 cm 2 , how accurately must the engineer cut on the side, assuming all sides have the same length? How do these numbers relate to δ , ε , a , and L ? 200. Use the precise definition of limit to prove that the following limit does not exist: lim x →1 | x −1| x −1 .
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