Calculus Examples

Evaluate the Limit limit as x approaches 0 of (sin(x^2))/x
limx0sin(x2)x
Step 1
Apply L'Hospital's rule.
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Step 1.1
Evaluate the limit of the numerator and the limit of the denominator.
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Step 1.1.1
Take the limit of the numerator and the limit of the denominator.
limx0sin(x2)limx0x
Step 1.1.2
Evaluate the limit of the numerator.
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Step 1.1.2.1
Evaluate the limit.
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Step 1.1.2.1.1
Move the limit inside the trig function because sine is continuous.
sin(limx0x2)limx0x
Step 1.1.2.1.2
Move the exponent 2 from x2 outside the limit using the Limits Power Rule.
sin((limx0x)2)limx0x
sin((limx0x)2)limx0x
Step 1.1.2.2
Evaluate the limit of x by plugging in 0 for x.
sin(02)limx0x
Step 1.1.2.3
Simplify the answer.
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Step 1.1.2.3.1
Raising 0 to any positive power yields 0.
sin(0)limx0x
Step 1.1.2.3.2
The exact value of sin(0) is 0.
0limx0x
0limx0x
0limx0x
Step 1.1.3
Evaluate the limit of x by plugging in 0 for x.
00
Step 1.1.4
The expression contains a division by 0. The expression is undefined.
Undefined
00
Step 1.2
Since 00 is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.
limx0sin(x2)x=limx0ddx[sin(x2)]ddx[x]
Step 1.3
Find the derivative of the numerator and denominator.
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Step 1.3.1
Differentiate the numerator and denominator.
limx0ddx[sin(x2)]ddx[x]
Step 1.3.2
Differentiate using the chain rule, which states that ddx[f(g(x))] is f(g(x))g(x) where f(x)=sin(x) and g(x)=x2.
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Step 1.3.2.1
To apply the Chain Rule, set u as x2.
limx0ddu[sin(u)]ddx[x2]ddx[x]
Step 1.3.2.2
The derivative of sin(u) with respect to u is cos(u).
limx0cos(u)ddx[x2]ddx[x]
Step 1.3.2.3
Replace all occurrences of u with x2.
limx0cos(x2)ddx[x2]ddx[x]
limx0cos(x2)ddx[x2]ddx[x]
Step 1.3.3
Differentiate using the Power Rule which states that ddx[xn] is nxn-1 where n=2.
limx0cos(x2)(2x)ddx[x]
Step 1.3.4
Reorder the factors of cos(x2)(2x).
limx02xcos(x2)ddx[x]
Step 1.3.5
Differentiate using the Power Rule which states that ddx[xn] is nxn-1 where n=1.
limx02xcos(x2)1
limx02xcos(x2)1
Step 1.4
Divide 2xcos(x2) by 1.
limx02xcos(x2)
limx02xcos(x2)
Step 2
Evaluate the limit.
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Step 2.1
Move the term 2 outside of the limit because it is constant with respect to x.
2limx0xcos(x2)
Step 2.2
Split the limit using the Product of Limits Rule on the limit as x approaches 0.
2limx0xlimx0cos(x2)
Step 2.3
Move the limit inside the trig function because cosine is continuous.
2limx0xcos(limx0x2)
Step 2.4
Move the exponent 2 from x2 outside the limit using the Limits Power Rule.
2limx0xcos((limx0x)2)
2limx0xcos((limx0x)2)
Step 3
Evaluate the limits by plugging in 0 for all occurrences of x.
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Step 3.1
Evaluate the limit of x by plugging in 0 for x.
20cos((limx0x)2)
Step 3.2
Evaluate the limit of x by plugging in 0 for x.
20cos(02)
20cos(02)
Step 4
Simplify the answer.
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Step 4.1
Multiply 2 by 0.
0cos(02)
Step 4.2
Raising 0 to any positive power yields 0.
0cos(0)
Step 4.3
The exact value of cos(0) is 1.
01
Step 4.4
Multiply 0 by 1.
0
0
 [x2  12  π  xdx ]