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This is one of our many quizzes on Estimating limit values from graphs. Click on this to access more such competitive quizzes on Calculus.

Estimating limit values from graphs Quiz 2

mathematics

 Estimate the limit of the graph given below  $x=0$. <img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/1f55e665-e3a7-47ec-a180-9a55486d2aa46137725020289636440.png"/>

 Analyze the graph carefully and we see that at $x=0$ the function has a value that is greater than 0.3 but less than 0.4 .

  Clearly as $x\to 0$, the value of the graph is greater than $0.3$ but less than $0.4$. From the options the value that is greater than $0.3$ and less than $0.4$ is $0.33$.

 Estimate the limit of the graph given below at $x=1$. <img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/254fcc4d-6f3b-4227-9c07-f6275e676bc33512441480979284256.png"/>

 Analyze the graph and use the equation of the straight line to calculate the slope of the line and then evaluate the limit of the function.

  $\text{Clearly from graph as }x\to 1,f\left( x \right)\to 5.4\text{ or 5}\text{.5} \\ \text{To check the exact value,} \text{ write the equation of graph,} \text{ since it is straight line its equation is given by } \\ y=mx+c \\ P\left( 0,2,8 \right),Q\left( 1.2,3,4 \right)\text{ are the points that lie on the straight line}\text{.} \\ \text{From graph }c=2.8 \\ 3.4=m\left( 1.2 \right)+2.8 \\ m=0.5 \\ y=0.5\times 1+2.8 \\ y\approx 3.3 $

 What is the nature the graph given below <img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/9c3550a0-38ba-4f7f-bd1b-dc84e56d1dc13596667181978573516.png"/>

 Analyze the graph and check the value of x and y if the value are finite then it is said to be converging else for infinite value it is diverging

  From graph: Since $\underset{x\to {{0}^{+}}}{{\lim }}\,f\left( x \right)=+\infty  \\ \underset{x\to {{0}^{-}}}{{\lim }}\,f\left( x \right)= - \infty  \\ $ Since the function does not have a finite value and the value increases infinitely therefore the given function is said to be diverging.

 What is the product of $f\left( x \right)$when $x\to 0$and $g\left( x \right)$when $x\to 0$and $x\to -1$. <img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/3c3637dd-c5be-47de-b25c-ea6bbe6d68128814793270053484020.png"/>

 Analyze the graph carefully and determine the limits of the functions given in the question.

  $\text{From the graph, we can say that,} \\ \underset{x\to 0}{{\lim }}\,f\left( x \right)=\underset{x\to 0}{{\lim }}\,\dfrac{\sin 2x}{4x}=0.5 \\ \text{     }\Rightarrow \underset{x\to 0}{{\lim }}\,f\left( x \right)=0.5 \\ \text{Also, }  \underset{x\to 0}{{\lim }}\,g\left( x \right)=\underset{x\to 0}{{\lim }}\,{{2}^{3x+2}}=4 \\ \text{     }\Rightarrow \underset{x\to 0}{{\lim }}\,g\left( x \right)=4 \\ \text{And, } \underset{x\to -1}{{\lim }}\,g\left( x \right)=\underset{x\to 0}{{\lim }}\,{{2}^{3x+2}}=0.5 \\ \text{     }\Rightarrow \underset{x\to -1}{{\lim }}\,g\left( x \right)=0.5 \\ \therefore \text{ the product of the limits given will be } 0.5\times 4\times 0.5=1 \\ \Rightarrow \text{The product of the limits will be }1$.

 Which one out of the functions $\dfrac{\sin 6x}{3{{x}^{2}}}$and  $\sqrt{9{{x}^{2}}+x}-x$  . Which one of them has a limit at $x\to \infty $ and what is the limit?

 First find the limits of individual functions by using the limit properties and as we see that the second function does not have a finite limit therefore function has a finite limit.

  ${{f}_{1}}\left( x \right)=\dfrac{\sin 6x}{3{{x}^{2}}} \\ \underset{x\to \infty }{{\lim }}\,{{f}_{1}}\left( x \right)=\underset{x\to \infty }{{\lim }}\,\dfrac{\sin 6x}{3{{x}^{2}}} \\ \sin x\in \left( -1,1 \right),\text{but }{{x}^{2}}\to 0 \\ \therefore x\to 0 \\ {{f}_{2}}\left( x \right)=\sqrt{9{{x}^{2}}+x}-x \\ \underset{x\to \infty }{{\lim }}\,{{f}_{2}}\left( x \right)=\underset{x\to \infty }{{\lim }}\,\sqrt{9{{x}^{2}}+x}-x \\ \Rightarrow \infty  \\ \text{Clearly  }\dfrac{\sin 6x}{3{{x}^{2}}}\text{ has a limit as }x\to \infty$