Limit : Trigonometric Function
Evaluate limx→π4sinx−cosxx−π4.
Solution (1)
limx→π4sinx−cosxx−π4
= limx→π4(sinx−cosxx−π4×√22√22)
= limx→π4√22sinx−√22cosx√22(x−π4)
= limx→π4√22sinx−√22cosx√22(x−π4)
= limx→π4sinxcosπ4−cosxsinπ4√22(x−π4)
= √2lim(x−π4)→0sin(x−π4)(x−π4)
=√2×1
=√2
Solution (2)
Let t=x−π4 and hence x=t+π4
When x→π4, t→0.
limx→π4sinx−cosxx−π4
=limt→0sin(t+π4)−cos(t+π4)t
=limt→0sintcosπ4+costsinπ4−costcosπ4+sintcosπ4t
=limt→02sintcosπ4t
=limt→02sint(√22)t
=√2limt→0sintt
=√2×1
=√2
Solution (1)
limx→π4sinx−cosxx−π4
= limx→π4(sinx−cosxx−π4×√22√22)
= limx→π4√22sinx−√22cosx√22(x−π4)
= limx→π4√22sinx−√22cosx√22(x−π4)
= limx→π4sinxcosπ4−cosxsinπ4√22(x−π4)
= √2lim(x−π4)→0sin(x−π4)(x−π4)
=√2×1
=√2
Solution (2)
Let t=x−π4 and hence x=t+π4
When x→π4, t→0.
limx→π4sinx−cosxx−π4
=limt→0sin(t+π4)−cos(t+π4)t
=limt→0sintcosπ4+costsinπ4−costcosπ4+sintcosπ4t
=limt→02sintcosπ4t
=limt→02sint(√22)t
=√2limt→0sintt
=√2×1
=√2
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