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Chapter 1: Q. 57 (page 136)

$\lim_{x \to 0} \frac{\sin x}{\cos x}$

Short Answer

Expert verified

$\lim_{x \to 0} \frac{\sin x}{\cos x} = 0$

Step by step solution

Step 1. Given information

$\lim_{x \to 0} \frac{\sin x}{\cos x}$

Step 2. Plug in the values

$\lim_{x \to 0} \frac{\sin x}{\cos x} = \frac{\sin (0)}{\cos (0)}$

$\lim_{x \to 0} \frac{\sin x}{\cos x} = 0$

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Most popular questions from this chapter

For each limit statement in Exercises $41 - 44$ , use algebra to find $δ$ or $N$ in terms of $ε$ or $M$ , according to the appropriate formal limit definition.

$\lim_{x \to 1^{-}} \frac{1}{1 - x} = \infty$ , find $δ$ in terms of $M$ .

Each function in Exercises 9–12 is discontinuous at some value x = c. Describe the type of discontinuity and any one-sided continuity at x = c, and sketch a possible graph of f.

$\lim_{x \to 2^{-}} f (x) = - \infty, \lim_{x \to 2^{+}} f (x) = \infty, f (2) = 3 .$

What are punctured intervals, and why do we need to use them when discussing limits?

For each limit $\underset{x \to c}{l i m} f (x) = L$ in Exercises 43–54, use graphs and algebra to approximate the largest value of $δ$ such that if localid="1648023101818" $x \in (c - δ, c) \cup (c, c + δ) then f (x) \in (L - ε, L + ε)$

localid="1648023199049" role="math" $\underset{x \to - 1}{l i m} \frac{x^{2} - 2 x - 3}{x + 1} = - 4, ε = 1$

For each limit statement , use algebra to find δ > 0 in terms of $ε$ > 0 so that if 0 < |x − c| < δ, then | f(x) − L| < $ε$ .

$\lim_{x \to 1} (3 x + 8) = 11$

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limx→0sinxcosx

Short Answer

Step by step solution

Step 1. Given information

Step 2. Plug in the values

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Most popular questions from this chapter

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$\lim_{x \to 0} \frac{\sin x}{\cos x}$