Limit calculator with steps

What this page covers

• Limits at a point, including $\frac{0}{0}$ forms that need factoring or rationalizing first.
• Limits at infinity of polynomial, rational and exponential expressions.
• One-sided limits, and L'Hôpital's rule for $\frac{0}{0}$ and $\frac{\infty}{\infty}$ forms.
• Not covered here: limits of sequences (use Sequence convergence calculator) and derivatives as answers in their own right (use Derivative calculator with steps).

How to enter the problem

• Type lim, then the variable and the point it approaches, such as x→2, then the function. Write ∞ for infinity.
• Scan it. Fit the one problem in the frame.
• Choose a photo. Several problems in one photo are solved as a list.
• Then tap Solve.

Worked examples

Common mistakes

• Writing $\frac{0}{0}$, or 0, as the answer. The form $\frac{0}{0}$ only says the expression must be rewritten before the limit can be found.
• At infinity, comparing the wrong terms. $\lim_{x\to\infty} \frac{4x^2+1}{2x^2-5x}$ is 2, the ratio of the $x^2$ coefficients, not $-\frac{4}{5}$.
• Using the quotient rule inside L'Hôpital's rule. Differentiate the numerator and the denominator separately.
• Using L'Hôpital's rule when the form is not $\frac{0}{0}$ or $\frac{\infty}{\infty}$. For $\frac{x+1}{x}$ at $x = 1$, substitution gives 2, while the rule would wrongly give 1.

Checks, assumptions and limits

• Check numerically. For the runnable example, $\frac{\sin(0.003)}{0.001} \approx 2.9999955$, close to 3.
• Angles are in radians. In degrees, $\frac{\sin(3x)}{x}$ would approach $\frac{\pi}{60}$ instead.
• A two-sided limit exists only when the left and right limits agree. $\frac{1}{x}$ has different one-sided limits at 0, so its limit there does not exist.
• HexMath can make mistakes. Double check important steps.

Related tasks

• Derivative calculator with steps
• Sequence convergence calculator with steps
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