Evaluate the following limit ( IE do not

Answered question

2022-03-20

Evaluate the following limit ( IE do not use L'Hopital's rule).

 

lim𝑥→4 sin(𝑥2−4𝑥)  sin(𝑥2−10𝑥+24) .limx4 sin(x24x)  sin(x210x+24) .


 

Answer & Explanation

alenahelenash

alenahelenash

Expert2022-06-06Added 556 answers

limx4sin(x2-4x)sin(x2-10x+24)

Evaluate the limit of the numerator and the limit of the denominator.

Take the limit of the numerator and the limit of the denominator.

limx4sin(x2-4x)limx4sin(x2-10x+24)

Evaluate the limit of the numerator.

0limx4sin(x2-10x+24)

Evaluate the limit of the denominator.

00

The expression contains a division by 00. The expression is undefined.

Undefined

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.

limx4sin(x2-4x)sin(x2-10x+24)=limx4ddx[sin(x2-4x)]ddx[sin(x2-10x+24)]

Find the derivative of the numerator and denominator.

limx4cos(x2-4x)(2x-4)cos(x2-10x+24)(2x-10)

Cancel the common factor of 2x-4 and 2x-10.

Factor 2 out ofcos(x2-4x)(2x-4).

limx42(cos(x2-4x)(x-2))cos(x2-10x+24)(2x-10)

Cancel the common factors.

limx4cos(x2-4x)(x-2)cos(x2-10x+24)(x-5)

Split the limit using the Limits Quotient Rule on the limit as xx approaches 4.

limx4cos(x2-4x)(x-2)limx4cos(x2-10x+24)(x-5)

Split the limit using the Product of Limits Rule on the limit as x approaches 4.

limx4cos(x2-4x)limx4x-2limx4cos(x2-10x+24)(x-5)

Move the limit inside the trig function because cosine is continuous.

cos(limx4x2-4x)limx4x-2limx4cos(x2-10x+24)(x-5)

Split the limit using the Sum of Limits Rule on the limit as x approaches 4.

cos(limx4x2-limx44x)limx4x-2limx4cos(x2-10x+24)(x-5)

Move the exponent 2 from x2 outside the limit using the Limits Power Rule.

cos((limx4x)2-limx44x)limx4x-2limx4cos(x2-10x+24)(x-5)

Move the term 4 outside of the limit because it is constant with respect to x.

cos((limx4x)2-(4limx4x))limx4x-2limx4cos(x2-10x+24)(x-5)

Split the limit using the Sum of Limits Rule on the limit as x approaches 4.

cos((limx4x)2-14limx4x)(limx4x-limx42)limx4cos(x2-10x+24)(x-5)

Evaluate the limit of 2 which is constant as xx approaches 4.

cos((limx4x)2-14limx4x)(limx4x-12)limx4cos(x2-10x+24)(x-5)

Split the limit using the Product of Limits Rule on the limit as x approaches 4.

cos((limx4x)2-14limx4x)(limx4x-12)limx4cos(x2-10x+24)limx4x-5

Move the limit inside the trig function because cosine is continuous.

cos((limx4x)2-14limx4x)(limx4x-12)cos(limx4x2-10x+24)limx4x-5

Split the limit using the Sum of Limits Rule on the limit as xx approaches 44.

cos((limx4x)2-14limx4x)(limx4x-12)cos(limx4x2-limx410x+limx424)limx4x-5

Move the exponent 2 from x2 outside the limit using the Limits Power Rule.

cos((limx4x)2-14limx4x)(limx4x-12)cos((limx4x)2-limx410x+limx424)limx4x-5

Move the term 10 outside of the limit because it is constant with respect to x.

cos((limx4x)2-14limx4x)(limx4x-12)cos((limx4x)2-110limx4x+limx424)limx4x-5

Evaluate the limit of 24 which is constant as xx approaches 4.

cos((limx4x)2-14limx4x)(limx4x-12)cos((limx4x)2-110limx4x+24)limx4x-5

Split the limit using the Sum of Limits Rule on the limit as x approaches 4.

cos((limx4x)2-14limx4x)(limx4x-12)cos((limx4x)2-110limx4x+24)(limx4x-limx45)

Evaluate the limit of 5 which is constant as x approaches 4.

cos((limx4x)2-14limx4x)(limx4x-12)cos((limx4x)2-110limx4x+24)(limx4x-15)

Evaluate the limits by plugging in 4 for all occurrences of x.

Evaluate the limit of x by plugging in 4 for x.

cos(42-14limx4x)(limx4x-12)cos((limx4x)2-110limx4x+24)(limx4x-15)

Evaluate the limit of x by plugging in 4 for x.

cos(42-144)(limx4x-12)cos((limx4x)2-110limx4x+24)(limx4x-15)

Evaluate the limit of x by plugging in 4 for x.

cos(42-144)(4-12)cos((limx4x)2-110limx4x+24)(limx4x-15)

Evaluate the limit of x by plugging in 4 for x.

cos(42-144)(4-12)cos(42-110limx4x+24)(limx4x-15)

Evaluate the limit of x by plugging in 4 for x.

cos(42-144)(4-12)cos(42-1104+24)(limx4x-15)

Evaluate the limit of x by plugging in 4 for x.

cos(42-144)(4-12)cos(42-1104+24)(4-15)

Simplify the numerator.

2cos(42-1104+24)(4-15)

Simplify the denominator.

2-1

Move the negative one from the denominator of 2-1.

-12

Multiply -1 by 2.

2

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