题目
题目

MAT135H5_F25_ALL SECTIONS 2.3 Preparation Check

多重下拉选择题

Which of the following statements must be true, and which are false? Note: each part is independent from the others. a) If 𝑓 ( 𝑥 ) is a polynomial, then lim 𝑥 → 5 𝑓 ( 𝑥 )  can be evaluated by computing 𝑓 ( 5 ) . [ Select ] False True b) If 𝑅 ( 𝑥 )   is a rational function, then lim 𝑥 → 7 𝑅 ( 𝑥 )     can be evaluated by computing 𝑅 ( 7 )  . [ Select ] False True c) If 𝑝 ( 𝑥 )  is a polynomial, then lim 𝑥 → 5 + 𝑝 ( 𝑥 )    can be evaluated by computing 𝑝 ( 5 )  . [ Select ] True False d) If 𝑓 ( 2 ) = 4  then lim 𝑥 → 2 [ 𝑓 ( 𝑥 ) ] 2 = 16   [ Select ] False True e) If   lim 𝑥 → 4 ( 5 𝑓 ( 𝑥 ) ) = 15   , then lim 𝑥 → 4 𝑓 ( 𝑥 ) = 3   . [ Select ] True False f) If   lim 𝑥 → 4 ( 𝑥 𝑓 ( 𝑥 ) ) = 8   , then lim 𝑥 → 4 𝑓 ( 𝑥 ) = 2    . [ Select ] False True

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思路分析
Here we go through each statement a–f and assess its truth value, keeping in mind basic limits rules for polynomials, rational functions, and general limit algebra. a) 'If f(x) is a polynomial, then lim_{x->5} f(x) can be evaluated by computing f(5).' - A polynomial is continuous at every real number, so the limit as x approaches 5 equals the function value at 5. In short, lim_{x->5} f(x) = f(5). This makes the statement true. b) 'If R(x) is a rational function,......Login to view full explanation

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类似问题

  the graph of f(x)f\left(x\right) given above. Use it to find the following one-sided and two-sided limits. (If a limit does not exist, write DNE.) limx→1−f(x)=\lim\limits_{x\:\rightarrow1^-}\:f\left(x\right)= [Fill in the blank], limx→1+f(x)=\lim\limits_{x\rightarrow1^+}f\left(x\right)= [Fill in the blank], limx→1f(x)=\lim\limits_{x\rightarrow1}f\left(x\right)\:= [Fill in the blank], limx→2f(x)=\lim\limits_{x\rightarrow2}f\left(x\right)= [Fill in the blank], limx→3−f(x)=\lim\limits_{x\rightarrow3^-}f\left(x\right)= [Fill in the blank], limx→3f(x)=\lim\limits_{x\rightarrow3}f\left(x\right)=[Fill in the blank], limx→4f(x)=\lim\limits_{x\rightarrow4}f\left(x\right)=[Fill in the blank],

                  Consider the two graphs above. What are the following limits? (If a limit does not exist, write DNE.) limx→1f(x)=\lim\limits_{x\rightarrow1}f\left(x\right)= [Fill in the blank], limx→1g(x)=\lim\limits_{x\rightarrow1}g\left(x\right)= [Fill in the blank], Note that the two functions f(x)f\left(x\right) and g(x)g\left(x\right) are identical except for at x=1x=1    . Is the following statement TRUE or FALSE? For any function  h(x)h\left(x\right) , the limit limx→ah(x)\lim\limits_{x\rightarrow a}h\left(x\right) does not depend on the value of h(x)h\left(x\right) at  x=ax=a  , or even whether h(a)h\left(a\right) is defined or not.  [Fill in the blank], (Write "TRUE" or "FALSE".)  

Question text Consider the function [math: f(x)={3x+5,x<33x2+4x−2,x≥3] f(x)= \begin{cases} \displaystyle & {3\,x+5}, & x < {3} \\ & {3\,x^2+4\,x-2}, & x \geq {3}\end{cases} . a) [math: limx→3−f(x)=]\displaystyle \lim_{{x \to {3}^-}} f(x) = [input] b) [math: limx→3+f(x)=]\displaystyle\lim_{{x \to {3}^+}} f(x) = [input] c) [math: limx→3f(x)=]\displaystyle\lim_{{x \to {3}}} f(x) = [select: (Clear my choice), does not exist since left limit is not equal to right limit., exists and equals 37] Check Question 3

Question text Consider the graph of [math: f(x)] shown below: a) [math: limx→2−f(x)=]\displaystyle \lim_{{x \to {2}^-}} f(x) =[input] b) [math: limx→2+f(x)=]\displaystyle\lim_{{x \to {2}^+}} f(x) =[input] c) [math: limx→2f(x)=]\displaystyle\lim_{{x \to {2}}} f(x) =[select: (Clear my choice), does not exist., exists and equals 1] Check Question 1

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