lim(x→0) [21(e^x - 1) - ln(1 + 21x)] / [sin(21x) - 21x]
You asked:
Evaluate the expression: \(\lim_{x \to 0}{\frac{21 \cdot \left({e}^{x} - 1\right) - \ln\left( 1 + 21 x \right)}{\sin\left( 21 x \right) - 21 x}}\)
MathBot Answer:
The two-sided limit \(\lim_{x \to 0}{\frac{21 \cdot \left({e}^{x} - 1\right) - \ln\left( 1 + 21 x \right)}{\sin\left( 21 x \right) - 21 x}}\) does not exist because the directional limits from the right and the left result in different values.
Right limit:
\(\displaystyle\lim_{x \to 0^+}\left(\frac{21 \left(e^{x} - 1\right) - \log{\left(21 x + 1 \right)}}{- 21 x + \sin{\left(21 x \right)}}\right) = -\infty\)
Left limit:
\(\displaystyle\lim_{x \to 0^-}\left(\frac{21 \left(e^{x} - 1\right) - \log{\left(21 x + 1 \right)}}{- 21 x + \sin{\left(21 x \right)}}\right) = \infty\)