Understanding the Limit
To evaluate the limit of the expression as x approaches 0:
Lim x-->0 (x(2^x-1))/(1-cosx)
we need to analyze both the numerator and denominator.
Step 1: Evaluate the Numerator
- The numerator is x(2^x - 1).
- As x approaches 0, 2^x approaches 1, hence 2^x - 1 approaches 0.
- Therefore, the overall product x(2^x - 1) also approaches 0.
Step 2: Evaluate the Denominator
- The denominator is (1 - cosx).
- As x approaches 0, cosx approaches 1, thus 1 - cosx approaches 0.
Indeterminate Form
- Since both the numerator and denominator approach 0, we have an indeterminate form (0/0).
Step 3: Applying L'Hôpital's Rule
- To resolve the indeterminate form, we can apply L'Hôpital's Rule, which states that if the limit produces an indeterminate form, we can take derivatives of the numerator and denominator.
Taking Derivatives
- Derivative of the numerator:
- Using product rule, the derivative of x(2^x - 1) is (2^x ln(2)) + (2^x - 1).
- Derivative of the denominator:
- The derivative of (1 - cosx) is sinx.
Evaluating the Limit Again
- Now, we re-evaluate the limit with the derivatives:
- Lim x-->0 ((2^x ln(2) + 2^x - 1)/sinx).
Final Step: Substitute x=0
- Substituting x=0 gives:
- The numerator becomes (0 * ln(2) + 0) = 0.
- The denominator becomes sin(0) = 0.
This still yields an indeterminate form, so we may need to apply L'Hôpital's Rule again or simplify further.
This detailed stepwise approach leads to a clearer understanding of the limit evaluation process in calculus.