Limit Calculator
Calculus
Intro: Compute one-sided and two-sided limits. Detects $0/0$ or $\infty/\infty$ and shows L’Hôpital’s rule or algebraic simplification.
Worked example
- $\displaystyle \lim_{x\to 0} \frac{\sin x}{x}$
- Identify the form: direct substitution gives $\sin 0 / 0 = 0/0$ (indeterminate).
- Apply L’Hôpital’s Rule (numerator and denominator differentiable near $0$): differentiate top and bottom.
- Derivative of $\sin x$ is $\cos x$; derivative of $x$ is $1$.
- Thus $\displaystyle \lim_{x\to 0} \frac{\sin x}{x} = \lim_{x\to 0} \frac{\cos x}{1} = \cos 0 = 1$.
- Conclusion: $\boxed{1}$.
- $\displaystyle \lim_{x\to 2} \frac{x^2-4}{x-2}$
- Identify the form: substitute $x=2$ to get $(4-4)/(2-2)=0/0$ (indeterminate).
- Factor the numerator using difference of squares: $x^2-4=(x-2)(x+2)$.
- Rewrite the fraction: $\displaystyle \frac{(x-2)(x+2)}{x-2}$.
- For $x\ne 2$, cancel the common factor $(x-2)$: the expression simplifies to $x+2$.
- Now take the limit of the simplified expression (a continuous function): $\lim_{x\to 2}(x+2)=2+2=4$.
- Conclusion: $\boxed{4}$.
- $\displaystyle \lim_{x\to 0} \frac{e^x-1}{x}$
- Identify the form: substitute $x=0$ to get $(e^0-1)/0 = 0/0$ (indeterminate).
- Apply L’Hôpital’s Rule: differentiate numerator and denominator.
- Derivative of $e^x$ is $e^x$; derivative of $x$ is $1$.
- Evaluate the new limit: $\displaystyle \lim_{x\to 0} \frac{e^x}{1} = e^0 = 1$.
- Conclusion: $\boxed{1}$.
FAQs
Does it use L'Hôpital automatically?
Yes—when an indeterminate form is detected.
Supports one-sided limits?
Yes—use x->a^+ or x->a^-.
Why choose Mathix?
- Get clear, step-by-step solutions that explain the “why,” not just the answer.
- See the rules used at each step (power, product, quotient, chain, and more).
- Optional animated walk-throughs to make tricky ideas click faster.
- Clean LaTeX rendering for notes, homework, and study guides.
Evaluate lim_{x->2} (x^2-4)/(x-2)
- (x^2-4) = (x-2)(x+2)
- For x != 2, cancel the common factor (x-2)
- Take the limit of x+2 as x->2 to get 4