Direct Substitution Method

Calculus 1 · Axiom Academy

EXAMPLE Direct Substitution Method Evaluating a limit by plugging in the point — valid exactly when the function is continuous there. Nice work. Because x^2 - 2x + 5 is a polynomial, it is continuous at x = 3 , so the limit is just the value of the function there. Why it works: if f is continuous at a , then — you may evaluate the limit by plugging in. Polynomials are continuous everywhere, so direct substitution always works for them. Plugging in can produce something undefined. Consider: This is an indeterminate form : direct substitution does not apply, so the value it shows is not the limit. You need another technique — factoring, multiplying by a conjugate, a known special limit, or L'Hôpital's Rule. (The true value of this particular limit is 1 .) Quick guide to what direct substitution returns: The rule of thumb: try plugging in first. If you get a clean number, you're done; if you get an indeterminate form, switch methods.

This is the written version of the interactive lesson above. See the full Calculus 1 course.