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Bài 23. Tìm các giới hạn sau :
a. \(\mathop {\lim }\limits_{x \to 2} \left( {3{x^2} + 7x + 11} \right)\)
b. \(\mathop {\lim }\limits_{x \to 1} {{x – {x^3}} \over {\left( {2x – 1} \right)\left( {{x^4} – 3} \right)}}\)
c. \(\mathop {\lim }\limits_{x \to 0} x\left( {1 – {1 \over x}} \right)\)
d. \(\mathop {\lim }\limits_{x \to 9} {{\sqrt x – 3} \over {9x – {x^2}}}\)
e. \(\mathop {\lim }\limits_{x \to \sqrt 3 } \left| {{x^2} – 4} \right|\)
f. \(\mathop {\lim }\limits_{x \to 2} \sqrt {{{{x^4} + 3x – 1} \over {2{x^2} – 1}}} \)
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a. \(\eqalign{& \mathop {\lim }\limits_{x \to 2} \left( {3{x^2} + 7x + 11} \right) = \mathop {\lim }\limits_{x \to 2} 3{x^2} + \mathop {\lim }\limits_{x \to 2} 7x + \mathop {\lim }\limits_{x \to 2} 11 \cr & = {3.2^2} + 7.2 + 11 = 37 \cr} \)
b. \(\mathop {\lim }\limits_{x \to 1} {{x – {x^3}} \over {\left( {2x – 1} \right)\left( {{x^4} – 3} \right)}} = {0 \over { – 2}} = 0\)
c. \(\mathop {\lim }\limits_{x \to 0} x\left( {1 – {1 \over x}} \right) = \mathop {\lim }\limits_{x \to 0} \left( {x – 1} \right) = – 1\)
d. \(\mathop {\lim }\limits_{x \to 9} {{\sqrt x – 3} \over {9x – {x^2}}} = \mathop {\lim }\limits_{x \to 9} {{\sqrt x – 3} \over { – x\left( {x – 9} \right)}} = – \mathop {\lim }\limits_{x \to 9} {1 \over {x\left( {\sqrt x + 3} \right)}} = – {1 \over {54}}\)
e. \(\mathop {\lim }\limits_{x \to \sqrt 3 } \left| {{x^2} – 4} \right| = 1\)
f. \(\mathop {\lim }\limits_{x \to 2} \sqrt {{{{x^4} + 3x – 1} \over {2{x^2} – 1}}} = \sqrt {{{{2^4} + 3.2 – 1} \over {{{22}^2} – 1}}} = \sqrt 3 \)