Evaluate limits involving the expressions (sin t/t), (1 - cos t)/t) and ((e ^ t- 1)/t) and indeterminate forms type "0/0"
Directions: Create example of limits of functions and evaluate their limits.
1.(sint)/t
2.(1 - cos t)/t
3.(e ^ t - 1)/t
Using L'Hospital's rule we can write:
limt→0sintt=limt→0(sint)’t’=limt→0cost1=1\displaystyle\lim_{\mathclap{t\to0}} {\frac{\sin t}{t}}= \lim_{\mathclap{t\to0}}{\frac{(\sin t)’}{t’}}= \lim_{\mathclap{t\to0}} {\frac{\cos t}{1}}=1t→0limtsint=t→0limt’(sint)’=t→0lim1cost=1
limt→01−costt=limt→0(1−cost)’t’=limt→0sint1=0\displaystyle\lim_{\mathclap{t\to0}} {\frac{1-\cos t}{t}}= \lim_{\mathclap{t\to0}}{\frac{(1-\cos t)’}{t’}}= \lim_{\mathclap{t\to0}} {\frac{\sin t}{1}}=0t→0limt1−cost=t→0limt’(1−cost)’=t→0lim1sint=0
limt→0et−1t=limt→0(et−1)’t’=limt→0et1=1\displaystyle\lim_{\mathclap{t\to0}} {\frac{e^t-1}{t}}= \lim_{\mathclap{t\to0}}{\frac{(e^t-1)’}{t’}}= \lim_{\mathclap{t\to0}} {\frac{e^t}{1}}=1t→0limtet−1=t→0limt’(et−1)’=t→0lim1et=1
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