CEGEP Calculus I — Limits with Trig, Exponential and Log Functions

All 8 questions

Each question targets one named concept from the sheet. Read them here, or print the PDF — it has working space under each one.

1. Q1Limits of Trigonometric Functions and the Squeeze Theorem

   Each limit gives ${\displaystyle \frac{0}{0}}$ on direct substitution. Rewrite the quotient with a trigonometric identity, cancel, and then substitute.

   1. ${\displaystyle {lim}_{x\to 0}\frac{1-\cos x}{{\sin }^{2}x}}$
   2. ${\displaystyle {lim}_{x\to \pi /2}\frac{\sin 2x}{\cos x}}$
   3. ${\displaystyle {lim}_{x\to 3\pi /4}\frac{1+\tan x}{\sin x+\cos x}}$
2. Q2Limits of Trigonometric Functions and the Squeeze Theorem

   Use the Squeeze Theorem for each limit. Write the inequality you squeeze with, and say why it holds.

   1. ${\displaystyle {lim}_{x\to 0}{x}^2(2-\cos \frac{1}{x})}$
   2. ${\displaystyle {lim}_{x\to 0^{+}}\sqrt{x}{e}^{\sin (1/x)}}$
3. Q3Limits of Exponential and Logarithmic Functions

   Find each limit. Where the answer is $\infty$ or $-\infty$, say which, and decide it from the sign of the quantity that approaches $0$.

   1. ${\displaystyle {lim}_{x\to 2^{-}}{e}^{1/(4-x^2)}}$ and ${\displaystyle {lim}_{x\to 2^{+}}{e}^{1/(4-x^2)}}$
   2. Is the line $x=2$ a vertical asymptote of $y=e^{1/(4-x^2)}$? Is $x=-2$?
   3. ${\displaystyle {lim}_{x\to 0^{+}}\ln (\sin x)}$
   4. ${\displaystyle {lim}_{x\to 1^{-}}\frac{2}{1-e^{x-1}}}$
4. Q4Limits of Exponential and Logarithmic Functions

   Each limit is indeterminate on direct substitution. Use a law of logarithms or factor an expression in $e^x$ (or in $3^x$), then evaluate.

   1. ${\displaystyle {lim}_{x\to 3^{+}}[\ln (x^2-9)-\ln (2x-6)]}$
   2. ${\displaystyle {lim}_{x\to \ln 2}\frac{e^{2x}-e^x-2}{e^x-2}}$
   3. ${\displaystyle {lim}_{x\to 0}\frac{{27}^x-1}{9^x-1}}$
   4. ${\displaystyle {lim}_{x\to 1}\frac{{\log }_3\left(x^4\right)}{{\log }_{9}x}}$
5. Q5Limits at Infinity with Exponentials, Logarithms and Arctangent

   Some of these limits exist and some do not. Evaluate each one that exists, and explain why each of the others does not.

   1. ${\displaystyle {lim}_{x\to -\infty }\cos \left(e^x\right)}$
   2. ${\displaystyle {lim}_{x\to \infty }\cos \left(e^x\right)}$
   3. ${\displaystyle {lim}_{x\to \infty }\arctan (x^2-5x)}$
   4. ${\displaystyle {lim}_{x\to \infty }\frac{5}{3-\ln x}}$
6. Q6Limits at Infinity with Exponentials, Logarithms and Arctangent

   Evaluate each limit. For a quotient of exponentials, divide the numerator and the denominator by the exponential that dominates in the direction $x$ is going.

   1. ${\displaystyle {lim}_{x\to \infty }\frac{4e^{2x}-e^x}{3+2e^{2x}}}$
   2. ${\displaystyle {lim}_{x\to \infty }\frac{5^x-4^x}{5^{x+1}+2·4^x}}$
   3. ${\displaystyle {lim}_{x\to -\infty }\frac{4e^{-2x}+1}{3-e^{-2x}}}$
   4. ${\displaystyle {lim}_{x\to \infty }[\ln (2x+1)-\frac{1}{2}\ln (x^2+3)]}$
7. Q7Continuity of Trigonometric, Exponential and Logarithmic Functions

   Find the largest intervals on which each function is continuous. For each one, name the facts about continuous functions that justify your answer.

   1. $f(x)=\arcsin (\ln x)$
   2. $g(x)={\displaystyle \frac{\sqrt{x+2}}{e^x-1}}$
   3. $h(x)=\ln (\cos x)$, for $-2\pi <x<2\pi$
8. Q8Synthesis — drawing on several topics in this set

   Let $$f(x)=\{\begin{array}{cc}{\displaystyle \frac{e^{2x}-e^x-6}{e^x-3}},\hfill & x<\ln 3,\hfill \\ k+6e^{-x},\hfill & x\ge \ln 3,\hfill \end{array}$$ where $k$ is a constant.

   1. Find the value of $k$ that makes $f$ continuous at $x=\ln 3$.
   2. With that value of $k$, find ${\displaystyle {lim}_{x\to -\infty }f(x)}$ and ${\displaystyle {lim}_{x\to \infty }f(x)}$, and state the horizontal asymptotes of $f$.
   3. Explain why $f$ is then continuous at every real number.

The 5 challenge problems for this topic are a separate, paid sheet and are not reproduced here.

The rest of this unit

The worksheet is the practice, and it is free. Two more printable documents cover the same unit and come with the CEGEP Calculus I bundle: read the notes first, work this sheet, then sit the test closed-book. See what each one covers.

What the Limits with Trig, Exponential and Log Functions notes cover  About the Limits with Trig, Exponential and Log Functions unit test

How to do every concept on this sheet

This is the part worksheet sites usually leave out. Below is the actual reasoning behind each group of questions — not a full solution set, the decisions that get you to one. Read it before you start, or after you get stuck.

A trigonometric 0/0 is settled by an identity

Q1 gives three quotients that all return 0/0 when you substitute. There is no polynomial to factor, but the same idea applies: rewrite the expression so the factor that is going to 0 appears in both the numerator and the denominator, cancel it, and substitute into what is left.

The Squeeze Theorem, when one factor has no limit at all

In Q2 the product law is unavailable, because cos(1/x) and sin(1/x) oscillate faster and faster near 0 and never approach a single value. What they do have is a bound, and a bound is all the Squeeze Theorem needs.

Exponentials and logarithms near a problem point

Q3 is about the sign of the quantity that is going to 0. In e^(1/(4 − x²)), the exponent grows without bound on one side of 2 and goes to −∞ on the other, so the same expression goes to ∞ from one side and to 0 from the other. A vertical asymptote needs an infinite limit from only one side, which is why part (b) asks you to decide it rather than assume it.

Q4 then gives limits that are indeterminate only on the surface. A difference of two logarithms that both go to −∞ becomes one logarithm by a log law, an expression in e^(2x) and e^x factors like a quadratic once you write u = e^x, and powers of 27 and 9 are powers of 3.

At infinity: bounded, levelling off, or never settling

Q5 puts the three behaviours side by side. When the inside of a continuous function settles, the function settles too, as with cos(e^x) as x → −∞. When the inside grows without bound, the answer depends on the outer function: arctan levels off at π/2, while cosine keeps returning to 1 and −1 and has no limit.

Continuity on the domain of a composition

Q7 asks for the largest intervals of continuity. The facts are short: each of these families is continuous wherever it is defined, a quotient is continuous where its denominator is not 0, and a composition is continuous wherever it is defined. So the whole question is the domain. For arcsin(ln x) the inside must lie between −1 and 1; for ln(cos x) the cosine must be positive. Answer with intervals, and close a bracket only where the function is defined at the endpoint.

The synthesis question

Q8 is a two-piece function with a constant to find. The left piece gives 0/0 at x = ln 3 and factors once you write u = e^x; forcing its limit to equal the right piece's value gives the constant. The two limits at infinity then come from the two pieces separately, one at each end, and give the horizontal asymptotes.

Getting the most out of it

Substitute first, then name the tool

Substituting tells you whether you are already finished, whether the form is 0/0, or whether a non-zero number over 0 makes the answer infinite. Only then choose: an identity, a log law, a substitution such as u = e^x, or the Squeeze Theorem.

Take the sides seriously

e^(1/x), ln near 0 and tan near π/2 all behave differently on the two sides of the point. Whenever the inside of the function is going to 0 or to ±∞, work out its sign first.

Justify continuity when you move a limit inside

Writing lim cos(e^x) = cos(lim e^x) is allowed because cosine is continuous at the inner limit. Say so in a clause. That clause is the reasoning a CEGEP marker is looking for.

Common questions