178
Chapter 2 | Limits
lim x →−7 ⎛ lim x →−9 ⎡
⎝ x · g ( x ) ⎞ ⎠
124.
⎣ x · f ( x )+2· g ( x ) ⎤ ⎦
125.
[T]
126.
True
or
False?
If
2 x −1≤ g ( x ) ≤ x 2 −2 x +3, then lim x →2
g ( x ) =0.
For the following problems, evaluate the limit using the squeeze theorem. Use a calculator to graph the functions f ( x ), g ( x ), and h ( x ) when possible.
⎛ ⎝ 1 θ
⎞ ⎠
θ 2 cos
127. [T] lim θ →0
⎧ ⎩ ⎨ 0, x rational x 2 , x irrrational
f ( x ), where f ( x ) =
128. lim x →0
129. [T] In physics, the magnitude of an electric field generated by a point charge at a distance r in vacuum is governed by Coulomb’s law: E ( r ) = q 4 πε 0 r 2 , where E represents the magnitude of the electric field, q is the charge of the particle, r is the distance between the particle and where the strength of the field is measured, and 1 4 πε 0 is Coulomb’s constant: 8.988×10 9 N·m 2 /C 2 . a. Use a graphing calculator to graph E ( r ) given that the charge of the particle is q =10 −10 . b. Evaluate lim r →0 + E ( r ). What is the physical meaning of this quantity? Is it physically relevant? Why are you evaluating from the right? 130. [T] The density of an object is given by its mass divided by its volume: ρ = m / V . a. Use a calculator to plot the volume as a function of density ⎛ ⎝ V = m / ρ ⎞ ⎠ , assuming you are examining something of mass 8 kg ( m =8). b. Evaluate lim ρ →0 + V ( ρ ) and explain the physical meaning.
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