ExploreCalc01HTML.html

[Maple Plot]

f(x) = PIECEWISE([x, 0 &lt; x and x &lt; 1],[x^2, 1 &lt; x and x &lt; 2],[1, 2 &lt; x and x &lt;= 3],[undefined, otherwise])

`Limiting value that f(x) approaches as x approaches 0 from the left:`

Limit(f(x),x = 0,left) = undefined, f(0) = undefined

`Limiting value that f(x) approaches as x approaches 0 from the right:`

Limit(f(x),x = 0,right) = 0, f(0) = undefined

`Limiting value that f(x) approaches as x approaches 0 from either side:`

Limit(f(x),x = 0) = undefined, f(0) = undefined

`Limiting value that f(x) approaches as x approaches 1 from the left:`

Limit(f(x),x = 1,left) = 1, f(1) = undefined

`Limiting value that f(x) approaches as x approaches 1 from the right:`

Limit(f(x),x = 1,right) = 1, f(1) = undefined

`Limiting value that f(x) approaches as x approaches 1 from either side:`

Limit(f(x),x = 1) = 1, f(1) = undefined

[Maple Plot]

f(x) = PIECEWISE([x, 0 &lt; x and x &lt; 1],[x^2, 1 &lt; x and x &lt; 2],[1, 2 &lt; x and x &lt;= 3],[undefined, otherwise])

`Limiting value that f(x) approaches as x approaches 2 from the left:`

Limit(f(x),x = 2,left) = 4, f(2) = undefined

`Limiting value that f(x) approaches as x approaches 2 from the right:`

Limit(f(x),x = 2,right) = 1, f(2) = undefined

`Limiting value that f(x) approaches as x approaches 2 from either side:`

Limit(f(x),x = 2) = undefined, f(2) = undefined

`Limiting value that f(x) approaches as x approaches 3 from the left:`

Limit(f(x),x = 3,left) = 1, f(3) = 1

`Limiting value that f(x) approaches as x approaches 3 from the right:`

Limit(f(x),x = 3,right) = undefined, f(3) = 1

`Limiting value that f(x) approaches as x approaches 3 from either side:`

Limit(f(x),x = 3) = undefined, f(3) = 1

[Maple Plot]

f(x) = PIECEWISE([1, x = 0],[x, 0 &lt; x and x &lt; 1],[2, x = 1],[x^2, 1 &lt; x and x &lt; 2],[3, x = 2],[1, 2 &lt; x and x &lt;= 3],[undefined, otherwise])

`Limiting value that f(x) approaches as x approaches 0 from the left:`

Limit(f(x),x = 0,left) = undefined, f(0) = 1

`Limiting value that f(x) approaches as x approaches 0 from the right:`

Limit(f(x),x = 0,right) = 0, f(0) = 1

`Limiting value that f(x) approaches as x approaches 0 from either side:`

Limit(f(x),x = 0) = undefined, f(0) = 1

`Limiting value that f(x) approaches as x approaches 1 from the left:`

Limit(f(x),x = 1,left) = 1, f(1) = 2

`Limiting value that f(x) approaches as x approaches 1 from the right:`

Limit(f(x),x = 1,right) = 1, f(1) = 2

`Limiting value that f(x) approaches as x approaches 1 from either side:`

Limit(f(x),x = 1) = 1, f(1) = 2

[Maple Plot]

f(x) = PIECEWISE([1, x = 0],[x, 0 &lt; x and x &lt; 1],[2, x = 1],[x^2, 1 &lt; x and x &lt; 2],[3, x = 2],[1, 2 &lt; x and x &lt;= 3],[undefined, otherwise])

`Limiting value that f(x) approaches as x approaches 2 from the left:`

Limit(f(x),x = 2,left) = 4, f(2) = 3

`Limiting value that f(x) approaches as x approaches 2 from the right:`

Limit(f(x),x = 2,right) = 1, f(2) = 3

`Limiting value that f(x) approaches as x approaches 2 from either side:`

Limit(f(x),x = 2) = undefined, f(2) = 3

`Limiting value that f(x) approaches as x approaches 3 from the left:`

Limit(f(x),x = 3,left) = 1, f(3) = 1

`Limiting value that f(x) approaches as x approaches 3 from the right:`

Limit(f(x),x = 3,right) = undefined, f(3) = 1

`Limiting value that f(x) approaches as x approaches 3 from either side:`

Limit(f(x),x = 3) = undefined, f(3) = 1

[Maple Plot]

f(x) = PIECEWISE([4-x, x &lt; 2],[1, x = 2],[x^2-2, 2 &lt; x])

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`Compute the following limits and fill in the blanks:`

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Limit(f(x),x = 0,left) = `____ `, Limit(f(x),x = 0,right) = `____ `, Limit(f(x),x = 0) = `____ `, f(0) = `____ `

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Limit(f(x),x = 1,left) = `____ `, Limit(f(x),x = 1,right) = `____ `, Limit(f(x),x = 1) = `____ `, f(1) = `____ `

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Limit(f(x),x = 2,left) = `____ `, Limit(f(x),x = 2,right) = `____ `, Limit(f(x),x = 2) = `____ `, f(2) = `____ `

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`Redefine the function f so that it is continuous:`

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[Maple Plot]

f(x) = PIECEWISE([x^2+4, x &lt; 1],[5, x = 1],[x^3, 1 &lt; x])

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`Compute the following limits and fill in the blanks:`

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Limit(f(x),x = -1,left) = `____ `, Limit(f(x),x = -1,right) = `____ `, Limit(f(x),x = -1) = `____ `, f(-1) = ____

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Limit(f(x),x = 0,left) = `____ `, Limit(f(x),x = 0,right) = `____ `, Limit(f(x),x = 0) = `____ `, f(0) = ____

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Limit(f(x),x = 1,left) = `____ `, Limit(f(x),x = 1,right) = `____ `, Limit(f(x),x = 1) = `____ `, f(1) = ____

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Limit(f(x),x = 2,left) = `____ `, Limit(f(x),x = 2,right) = `____ `, Limit(f(x),x = 2) = `____ `, f(2) = ____

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`Redefine the function f so that it is continuous:`

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[Maple Plot]

f(x) = PIECEWISE([2*x+9, x &lt; -2],[x^2+1, -2 &lt; x and x &lt;= 1],[3*x-1, 1 &lt; x and x &lt; 3],[x+6, 3 &lt; x],[undefined, otherwise])

``

`Compute the following limits and fill in the blanks:`

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Limit(f(x),x = -2,left) = `____ `, Limit(f(x),x = -2,right) = `____ `, Limit(f(x),x = -2) = `____ `, f(-2) = ____

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Limit(f(x),x = -1,left) = `____ `, Limit(f(x),x = -1,right) = `____ `, Limit(f(x),x = -1) = `____ `, f(-1) = ____

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Limit(f(x),x = 0,left) = `____ `, Limit(f(x),x = 0,right) = `____ `, Limit(f(x),x = 0) = `____ `, f(0) = ____

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Limit(f(x),x = 1,left) = `____ `, Limit(f(x),x = 1,right) = `____ `, Limit(f(x),x = 1) = `____ `, f(1) = ____

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Limit(f(x),x = 2,left) = `____ `, Limit(f(x),x = 2,right) = `____ `, Limit(f(x),x = 2) = `____ `, f(2) = ____

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Limit(f(x),x = 3,left) = `____ `, Limit(f(x),x = 3,right) = `____ `, Limit(f(x),x = 3) = `____ `, f(3) = ____

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`Redefine the function f so that it is continuous:`

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`Left hand limit`, Limit(f(x),x = a,left) = b

`Right hand limit`, Limit(f(x),x = a,right) = b

`Two-sided limit`, Limit(f(x),x = a) = b

`The function `*f(x)*` is continuous at x ` = a