A limit asks what a function value approaches as x gets close to a number. Most limits are quick: you try substitution, and if that fails with a 0/0 form, you do one clean algebra move (factor/cancel or rationalize) and try again.
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You do not have to hit x = a exactly. The statement lim(x→a) f(x) = L means: when x gets close to a (from either side), f(x) gets close to L.
That’s why limits can exist even if the function is undefined at a point (like a hole after cancellation). Limits are about behavior near a point.
If you want a clean overview of the common limit techniques and what they mean graphically, OpenStax gives a friendly introduction in its section on limits of functions. The big win is learning the small set of patterns that appear over and over.
Problem: Evaluate lim(x→3) (x^2 + 2x − 1).
Polynomials are continuous everywhere, so substitution is valid:
Problem: Evaluate lim(x→1) (x^2 − 1)/(x − 1).
Substituting gives 0/0, so factor:
Now substitute into the simplified expression:
Interpretation: the original function has a hole at x = 1, but the values approach 2.
Problem: Evaluate lim(x→0) (√(x + 9) − 3)/x.
Substitution gives (3 − 3)/0 = 0/0. Rationalize by multiplying by the conjugate:
Now substitute x = 0:
A two-sided limit exists only if the left-hand limit and right-hand limit match:
Typical “one-sided” triggers: absolute value, piecewise functions, and rational functions with vertical asymptotes.
Problem: Evaluate lim(x→0) |x|/x.
For x > 0, |x| = x, so |x|/x = 1. For x < 0, |x| = −x, so |x|/x = −1.
Since the one-sided limits are different, the two-sided limit does not exist.
Sometimes substitution produces division by zero that does not cancel (like 5/0). In that case, the function may blow up to +∞ or −∞. The sign depends on the direction you approach and the signs of numerator/denominator.
Fast sign check:
The most important trig limit is:
Many trig limits are solved by rewriting the expression until you can use that pattern. For example:
Example: Evaluate lim(x→0) (sin(5x))/(x).
Multiply and divide by 5 to create sin(5x)/(5x):
The key idea is “make the denominator match what’s inside sine.”
Another common category is what happens as x → ∞ or x → −∞. For rational functions (polynomial over polynomial), the limit is controlled by the highest powers.
Example: Evaluate lim(x→∞) (2x^2 − x + 1)/(x^2 + 4).
Divide top and bottom by x^2:
Shortcut: if the degrees match, the limit is the ratio of leading coefficients.
0/0 means “try again after simplifying,” not “the limit is 0.” Factor/cancel or rationalize.
You can only cancel factors, not terms. For example, you can cancel (x − 1) if it is a factor, not if it’s part of a sum.
For absolute values and piecewise functions, you must check left and right. Matching is required for the limit to exist.
If both sides head to +∞, the limit is +∞ (an infinite limit). If one side is +∞ and the other is −∞, the two-sided limit DNE.
If you get 0/0, write the simplification move you used (factor, conjugate, or common denominator). That is usually what the teacher is grading.
Paste a limit and see the exact algebra move needed (or whether the limit DNE).
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