Các phương pháp tìm giới hạn

A flow chart has options A through H, as follows. Step A, direct substitution. Try to evaluate the function directly. Evaluating f of a leads to options B through D. Option B: f of a = start fraction b divided by 0 end fraction, where b is not zero. The result is asymptote (probably). Example: the limit of start fraction 1 divided by x minus 1 end fraction as x approaches 1. Inspect with a graph or table to learn more about the function at x = a. Option C: f of a = b, where b is a real number. The result is limit found (probably). Example: limit of x squared as x approaches 3 = 3 squared = 9. Option D: f of a = start fraction 0 divided by 0 end fraction. Result is indeterminate form. Example: limit of start fraction x squared minus x minus 2 divided by x squared minus 2 x minus 3 end fraction, as x approaches negative 1. If you obtained option D, try rewriting the limit in an equivalent form. This leads to options E through G. Option E: factoring. Example: limit of start fraction x squared minus x minus 2 divided by x squared minus 2 x minus 3 end fraction, as x approaches negative 1 can be reduced to the limit of start fraction x minus 2 divided by x minus 3 end fraction as x approaches negative 1, by factoring and cancelling. Option F: conjugates. Example: the limit of start fraction start square root x end square root minus 2 divided by x minus 4 end fraction as x approaches 4 can be rewritten as the limit of start fraction 1 divided by start square root x end square root + 2 end fraction as x approaches 4, using conjugates and cancelling. Option G: trig identities. Example: limit of start fraction sine of x divided by sine of 2 x end fraction as x approaches 0 can be rewritten as the limit of start fraction 1 divided by 2 côsin of x end fraction as x approaches 0, using a trig identity. Using options E through G, try evaluating the limit in its new form, circling back to A, direct substitution. The last option is H, approximation: when all else fails, graphs and tables can help approximate limits.