Evaluate the following limit: limit as x rightwards arrow 0 to the power of plus of x to the power of short dash 1 end exponent sin left parenthesis 8 x right parenthesis.A. limit as x rightwards arrow 0 to the power of plus of x to the power of short dash 1 end exponent sin left parenthesis 8 x right parenthesis equals short dash 8B. limit as x rightwards arrow 0 to the power of plus of x to the power of short dash 1 end exponent sin left parenthesis 8 x right parenthesis equals 8C. limit as x rightwards arrow 0 to the power of plus of x to the power of short dash 1 end exponent sin left parenthesis 8 x right parenthesis equals 1D. limit as x rightwards arrow 0 to the power of plus of x to the power of short dash 1 end exponent sin left parenthesis 8 x right parenthesis equals 0
Question
Evaluate the following limit: limit as x rightwards arrow 0 to the power of plus of x to the power of short dash 1 end exponent sin left parenthesis 8 x right parenthesis.A. limit as x rightwards arrow 0 to the power of plus of x to the power of short dash 1 end exponent sin left parenthesis 8 x right parenthesis equals short dash 8B. limit as x rightwards arrow 0 to the power of plus of x to the power of short dash 1 end exponent sin left parenthesis 8 x right parenthesis equals 8C. limit as x rightwards arrow 0 to the power of plus of x to the power of short dash 1 end exponent sin left parenthesis 8 x right parenthesis equals 1D. limit as x rightwards arrow 0 to the power of plus of x to the power of short dash 1 end exponent sin left parenthesis 8 x right parenthesis equals 0
Solution
The limit as x approaches 0 from the positive side of (x^-1)sin(8x) can be evaluated using L'Hopital's Rule, which states that the limit of a quotient of two functions as x approaches a certain value is equal to the limit of the quotients of their derivatives.
First, rewrite the expression as a quotient: (sin(8x))/x.
Then, take the derivative of the numerator and the denominator:
The derivative of sin(8x) with respect to x is 8cos(8x). The derivative of x with respect to x is 1.
So, the limit as x approaches 0 from the positive side of (x^-1)sin(8x) is equal to the limit as x approaches 0 from the positive side of (8cos(8x))/1.
As x approaches 0, cos(8x) approaches cos(0), which is 1. Therefore, the limit is 8.
So, the correct answer is B. limit as x rightwards arrow 0 to the power of plus of x to the power of short dash 1 end exponent sin left parenthesis 8 x right parenthesis equals 8.
Similar Questions
Given limit as x rightwards arrow short dash 1 of f left parenthesis x right parenthesis equals 13 and limit as x rightwards arrow short dash 1 of g left parenthesis x right parenthesis equals short dash 5, evaluate limit as x rightwards arrow short dash 1 of left square bracket f left parenthesis x right parenthesis g left parenthesis x right parenthesis minus 19 right square bracket.A. limit as x rightwards arrow short dash 1 of left square bracket f left parenthesis x right parenthesis g left parenthesis x right parenthesis minus 19 right square bracket equals short dash 46B. limit as x rightwards arrow short dash 1 of left square bracket f left parenthesis x right parenthesis g left parenthesis x right parenthesis minus 19 right square bracket equals short dash 64C. limit as x rightwards arrow short dash 1 of left square bracket f left parenthesis x right parenthesis g left parenthesis x right parenthesis minus 19 right square bracket equals 46D. limit as x rightwards arrow short dash 1 of left square bracket f left parenthesis x right parenthesis g left parenthesis x right parenthesis minus 19 right square bracket equals short dash 84
Evaluate limit as x rightwards arrow 5 of fraction numerator short dash 3 x plus 7 space over denominator open parentheses x minus 5 close parentheses squared end fraction numerically.A. 0B. Does not existC. short dash infinityD. infinity
YouEvaluate the following limit: limit as x rightwards arrow 0 of fraction numerator 2 cos left parenthesis 5 x right parenthesis minus 2 over denominator 3 x end fraction.A. limit as x rightwards arrow 0 of fraction numerator 2 cos left parenthesis 5 x right parenthesis minus 2 over denominator 3 x end fraction equals 10 over 3B. limit as x rightwards arrow 0 of fraction numerator 2 cos left parenthesis 5 x right parenthesis minus 2 over denominator 3 x end fraction equals 2 over 3C. limit as x rightwards arrow 0 of fraction numerator 2 cos left parenthesis 5 x right parenthesis minus 2 over denominator 3 x end fraction equals 8 over 3D. limit as x rightwards arrow 0 of fraction numerator 2 cos left parenthesis 5 x right parenthesis minus 2 over denominator 3 x end fraction equals 0
Evaluate the limit:lim𝑥→−5𝑥+5𝑥2−25=
Evaluate the expression limx−→∞ x2ex
Upgrade your grade with Knowee
Get personalized homework help. Review tough concepts in more detail, or go deeper into your topic by exploring other relevant questions.