Introduction to Calculus: Limit

Introduction to Calculus​

​Algebra and Calculus are two different disciplines. In algebra, we solve equations for a particular value of a variable. In Calculus we are interested in how a change in one variable affects other variables.

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​Illustrations and samples of limit

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​Limit of a Function

​}-describe how a function behaves as the independent variables get closer to a certain number.

}Can be analyzed either numerically , graphically or algebraically

}Generalized input-output process that defines a mapping of a set of input values to a set of output values.

Multiple Choice

Calculus is a branch of Mathematics that studies ______________ .

• Chance

• Change

• Choice

• None of the above

Limits

Limits

Multiple Choice

Evaluate

$$\lim_{x\rightarrow4}\ \frac{x^2\ +\ 16}{x\ -\ 4}$$  

• 8

• DNE

• 0

• -4

Multiple Choice

Evaluate

$$\lim_{x\rightarrow4}\ \frac{x^2\ -\ 16}{x\ -\ 4}$$ .

• 8

• 1

• 0

• DNE

Multiple Choice

What is the limit of the function as x approaches -4 from the left?

• 2

• -4

• DNE

• -2

Multiple Choice

Find f(2).

• 1

• -1

• 5

• DNE

Multiple Choice

Find the limit of the function as x approaches 2.

• 1

• -1

• 5

• DNE

Multiple Choice

Find the limit of the function as x approaches 2⁺.

• 1

• -1

• 5

• DNE

Multiple Choice

Evaluate 

• -7
• 0
• 1
• DNE

Multiple Choice

Find the limit as x approaches 1.

• 0

• 3

• DNE

• ∞

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Multiple Choice

Compare $$f\left(-2\right)\ and\ \lim\ as\ x\rightarrow-2$$

• $$f\left(-2\right)=0=\lim_{x\rightarrow-2}f\left(x\right)$$  
  

• $$f\left(-2\right)=0\ne\lim_{x\rightarrow-2}f\left(x\right)$$  

• $$f\left(-2\right)=-1=\lim_{x\rightarrow-2}f\left(x\right)$$  
  

• $$f\left(-2\right)=-1\ne\lim_{x\rightarrow-2}f\left(x\right)$$  
  

Multiple Choice

What is the limit of the function f(x) as it approaches to 1 given the table of values?

• -2

• -1

• 1

• 2

Multiple Choice

Find the limit.

• Does not exist

• $$\frac{3}{5}$$  

• 13

• 9

Multiple Choice

Determine $$\lim_{x\rightarrow-5}\ \frac{x+5}{x^2+7x+10}$$  

• -1/3

• 0

• 1/3

• undefined

Multiple Choice

Determine $$\lim_{x\rightarrow\ 1}\ln\left(x^2+x-1\right)$$  

• -1

• 0

• 1

• Does not exist

Multiple Choice

Determine $$\lim_{x\rightarrow\ \frac{\pi}{3}}\left(-\cos x\right)$$  

• -1/2

• 0

• 1/2

• 1

Multiple Choice

What is $$\lim_{x\rightarrow90\degree}\frac{4\sin x-3\cos x}{2\sin\ x}$$  

• 1

• 2

• 3

• 4

Multiple Choice

$$\lim_{x\rightarrow5}\ 3x^3\ +\ 2\ =\ \lim_{x\rightarrow5}\ 3x^3\ +\ \lim_{x\rightarrow5}\ 2$$  
Which of the following theorems justify the given equation?

• Limit of a function x

• Limit of a constant and function x

• Limit of sum and difference of  function

• limit of product of a function

Multiple Choice

Which of the following illustrates limit of a constant and a function?

• $$\lim_{x\rightarrow1}\ x^2\ =\ \left(\lim_{x\rightarrow1}\ x\right)^2$$

• $$\lim_{x\rightarrow2}-2\left(x+5\right)\ =\lim_{x\rightarrow2}\ -2\ \times\ \lim_{x\rightarrow2}\left(x+5\right)$$

• $$\lim_{x\rightarrow\ \frac{1}{2\ }}\ -3x\ =\ -3\ \left(\lim_{x\rightarrow\ \frac{1}{2\ }}x\right)$$

Multiple Choice

What is $$\lim_{x\rightarrow0}\frac{1}{x}$$  ?

• $$-1$$  

• $$1$$  

• $$0$$  

• $$\infty$$  

Multiple Choice

What is $$\lim_{x\rightarrow\infty}\frac{1}{x}$$  ?

• $$-1$$  

• $$1$$  

• $$0$$  

• $$\infty$$  

Multiple Choice

$$\lim_{x\rightarrow\infty}\ \frac{2x^3}{x^3}=\ ...$$  

• - 4

• - 2

• 1

• 2

• $$\infty$$  

Multiple Choice

$$\lim_{x\rightarrow\infty}\ \frac{1-2x+2x^3}{x^3+x+1}=\ ...$$  

• - 4

• - 2

• 1

• 2

• $$\infty$$  

Multiple Choice

$$\lim_{x\rightarrow\infty}\ \frac{x^4\ +3x^2+2}{x^3+x^2+1}\ =\ ...$$  

• 0

• 1

• 2

• 3

• $$\infty$$