SUMMARY
The limit problem presented is $$\lim_{{x}\to{0 }}\frac{\sqrt{ax+b}-2 }{x}=1$$, where the values of $a$ and $b$ must be determined. It is established that for the limit to exist, the expression must yield a 0/0 form, leading to the conclusion that $b = 4$. Applying L'Hôpital's rule, the derivative of the numerator at $x=0$ gives $\frac{a}{2\sqrt{4}}=1$, resulting in $a=4$. Thus, the definitive values are $a = 4$ and $b = 4$.
PREREQUISITES
- Understanding of limits in calculus
- Familiarity with L'Hôpital's rule
- Knowledge of derivatives and their application
- Basic algebraic manipulation skills
NEXT STEPS
- Study L'Hôpital's rule in depth
- Explore advanced limit techniques in calculus
- Review the properties of square roots in calculus
- Practice solving limit problems involving derivatives
USEFUL FOR
Students and educators in calculus, mathematicians focusing on limits, and anyone seeking to deepen their understanding of derivative applications in limit problems.